Theorem spcimgf

Description: Rule of specialization, using implicit substitution. Compare Theorem 7.3 of Quine p. 44. (Contributed by Mario Carneiro, 4-Jan-2017)

	Ref	Expression
Hypotheses	spcimgf.1	|- F/_ x A
	spcimgf.2	|- F/ x ps
	spcimgf.3	|- ( x = A -> ( ph -> ps ) )
Assertion	spcimgf	|- ( A e. V -> ( A. x ph -> ps ) )

Proof

Step	Hyp	Ref	Expression
1		spcimgf.1	|- F/_ x A
2		spcimgf.2	|- F/ x ps
3		spcimgf.3	|- ( x = A -> ( ph -> ps ) )
4	2 1	spcimgfi1	|- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A e. V -> ( A. x ph -> ps ) ) )
5	4 3	mpg	|- ( A e. V -> ( A. x ph -> ps ) )