Theorem exdistrf 2333

Description: Distribution of existential quantifiers, with a bound-variable hypothesis saying that y is not free in ph, but x can be free in ph (and there is no distinct variable condition on x and y). (Contributed by Mario Carneiro, 20-Mar-2013.) (Proof shortened by Wolf Lammen, 14-May-2018.)

Hypothesis

Ref	Expression
exdistrf.1	|- ( -. A. x x = y -> F/ y ph )

Assertion

Ref	Expression
exdistrf	|- ( E. x E. y ( ph /\ ps ) -> E. x ( ph /\ E. y ps ) )

Proof of Theorem exdistrf

Step	Hyp	Ref	Expression
1		nfe1 2027	. 2 |- F/ x E. x ( ph /\ E. y ps )
2		19.8a 2052	. . . . . 6 |- ( ps -> E. y ps )
3	2	anim2i 593	. . . . 5 |- ( ( ph /\ ps ) -> ( ph /\ E. y ps ) )
4	3	eximi 1762	. . . 4 |- ( E. y ( ph /\ ps ) -> E. y ( ph /\ E. y ps ) )
5		biidd 252	. . . . 5 |- ( A. x x = y -> ( ( ph /\ E. y ps ) <-> ( ph /\ E. y ps ) ) )
6	5	drex1 2327	. . . 4 |- ( A. x x = y -> ( E. x ( ph /\ E. y ps ) <-> E. y ( ph /\ E. y ps ) ) )
7	4, 6	syl5ibr 236	. . 3 |- ( A. x x = y -> ( E. y ( ph /\ ps ) -> E. x ( ph /\ E. y ps ) ) )
8		19.40 1797	. . . 4 |- ( E. y ( ph /\ ps ) -> ( E. y ph /\ E. y ps ) )
9		exdistrf.1	. . . . . 6 |- ( -. A. x x = y -> F/ y ph )
10	9	19.9d 2070	. . . . 5 |- ( -. A. x x = y -> ( E. y ph -> ph ) )
11	10	anim1d 588	. . . 4 |- ( -. A. x x = y -> ( ( E. y ph /\ E. y ps ) -> ( ph /\ E. y ps ) ) )
12		19.8a 2052	. . . 4 |- ( ( ph /\ E. y ps ) -> E. x ( ph /\ E. y ps ) )
13	8, 11, 12	syl56 36	. . 3 |- ( -. A. x x = y -> ( E. y ( ph /\ ps ) -> E. x ( ph /\ E. y ps ) ) )
14	7, 13	pm2.61i 176	. 2 |- ( E. y ( ph /\ ps ) -> E. x ( ph /\ E. y ps ) )
15	1, 14	exlimi 2086	1 |- ( E. x E. y ( ph /\ ps ) -> E. x ( ph /\ E. y ps ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   A.wal 1481   E.wex 1704   F/wnf 1708

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-12 2047  ax-13 2246

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

This theorem is referenced by:  oprabid  6677