Theorem 19.9d2rf 29318

Description: A deduction version of one direction of 19.9 2072 with two variables. (Contributed by Thierry Arnoux, 20-Mar-2017.)

Hypotheses

Ref	Expression
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

Ref	Expression
19.9d2rf	|- ( ph -> ps )

Proof of Theorem 19.9d2rf

Step	Hyp	Ref	Expression
1		19.9d2rf.3	. . . 4 |- ( ph -> E. x e. A E. y e. B ps )
2		rexex 3002	. . . 4 |- ( E. x e. A E. y e. B ps -> E. x E. y e. B ps )
3		rexex 3002	. . . . 5 |- ( E. y e. B ps -> E. y ps )
4	3	eximi 1762	. . . 4 |- ( E. x E. y e. B ps -> E. x E. y ps )
5	1, 2, 4	3syl 18	. . 3 |- ( ph -> E. x E. y ps )
6		19.9d2rf.0	. . . . 5 |- F/ y ph
7		19.9d2rf.1	. . . . 5 |- ( ph -> F/ x ps )
8	6, 7	nfexd 2167	. . . 4 |- ( ph -> F/ x E. y ps )
9	8	19.9d 2070	. . 3 |- ( ph -> ( E. x E. y ps -> E. y ps ) )
10	5, 9	mpd 15	. 2 |- ( ph -> E. y ps )
11		19.9d2rf.2	. . 3 |- ( ph -> F/ y ps )
12	11	19.9d 2070	. 2 |- ( ph -> ( E. y ps -> ps ) )
13	10, 12	mpd 15	1 |- ( ph -> ps )

Syntax hints:   -> wi 4   E.wex 1704   F/wnf 1708   E.wrex 2913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-rex 2918

This theorem is referenced by:  19.9d2r  29319  xrofsup  29533