Theorem spcgf

Description: Rule of specialization, using implicit substitution. Compare Theorem 7.3 of Quine p. 44. (Contributed by NM, 2-Feb-1997) (Revised by Andrew Salmon, 12-Aug-2011)

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

Proof

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