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  Objects: a, b, c
  Functions: f, g

1 EY:AX:p(X,Y) Premise 2 AX:p(X,d) Assumption 3 p(c,d) Universal Elimination 2 4 EY:p(c,Y) Existential Introduction 3 5 AX:p(X,d) => EY:p(c,Y) Implication Introduction 2 4 6 AY:(AX:p(X,Y) => EY:p(c,Y)) Universal Introduction 5 7 EY:p(c,Y) Existential Elimination 1 6 8 AX:EY:p(X,Y) Universal Introduction 7