Theorem fmptdf

Description: A version of fmptd using bound-variable hypothesis instead of a distinct variable condition for ph . (Contributed by Glauco Siliprandi, 29-Jun-2017)

	Ref	Expression
Hypotheses	fmptdf.1	|- F/ x ph
	fmptdf.2	|- ( ( ph /\ x e. A ) -> B e. C )
	fmptdf.3	|- F = ( x e. A |-> B )
Assertion	fmptdf	|- ( ph -> F : A --> C )

Proof

Step	Hyp	Ref	Expression
1		fmptdf.1	|- F/ x ph
2		fmptdf.2	|- ( ( ph /\ x e. A ) -> B e. C )
3		fmptdf.3	|- F = ( x e. A |-> B )
4	2	ex	|- ( ph -> ( x e. A -> B e. C ) )
5	1 4	ralrimi	|- ( ph -> A. x e. A B e. C )
6	3	fmpt	|- ( A. x e. A B e. C <-> F : A --> C )
7	5 6	sylib	|- ( ph -> F : A --> C )