Theorem 19.9d2rf

Description: A deduction version of one direction of 19.9 with two variables. (Contributed by Thierry Arnoux, 20-Mar-2017)

	Ref	Expression
Hypotheses	19.9d2rf.0	|- F/ y ph
	19.9d2rf.1	|- ( ph -> F/ x ps )
	19.9d2rf.2	|- ( ph -> F/ y ps )
	19.9d2rf.3	|- ( ph -> E. x e. A E. y e. B ps )
Assertion	19.9d2rf	|- ( ph -> ps )

Proof

Step	Hyp	Ref	Expression
1		19.9d2rf.0	|- F/ y ph
2		19.9d2rf.1	|- ( ph -> F/ x ps )
3		19.9d2rf.2	|- ( ph -> F/ y ps )
4		19.9d2rf.3	|- ( ph -> E. x e. A E. y e. B ps )
5		rexex	|- ( E. x e. A E. y e. B ps -> E. x E. y e. B ps )
6		rexex	|- ( E. y e. B ps -> E. y ps )
7	6	eximi	|- ( E. x E. y e. B ps -> E. x E. y ps )
8	4 5 7	3syl	|- ( ph -> E. x E. y ps )
9	1 2	nfexd	|- ( ph -> F/ x E. y ps )
10	9	19.9d	|- ( ph -> ( E. x E. y ps -> E. y ps ) )
11	8 10	mpd	|- ( ph -> E. y ps )
12	3	19.9d	|- ( ph -> ( E. y ps -> ps ) )
13	11 12	mpd	|- ( ph -> ps )