Theorem exdistrfor 1678

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 Jim Kingdon, 25-Feb-2018.)

Hypothesis

Ref	Expression
exdistrfor.1	|- (A.x x = y \/ A.x F/yph)

Assertion

Ref	Expression
exdistrfor	|- (E.xE.y(ph /\ ps) -> E.x(ph /\ E.yps))

Proof of Theorem exdistrfor

Step	Hyp	Ref	Expression
1		exdistrfor.1	. 2 |- (A.x x = y \/ A.x F/yph)
2		biidd 161	. . . . . 6 |- (A.x x = y -> ((ph /\ ps) <-> (ph /\ ps)))
3	2	drex1 1676	. . . . 5 |- (A.x x = y -> (E.x(ph /\ ps) <-> E.y(ph /\ ps)))
4	3	drex2 1617	. . . 4 |- (A.x x = y -> (E.xE.x(ph /\ ps) <-> E.xE.y(ph /\ ps)))
5		hbe1 1381	. . . . . 6 |- (E.x(ph /\ ps) -> A.xE.x(ph /\ ps))
6	5	19.9h 1531	. . . . 5 |- (E.xE.x(ph /\ ps) <-> E.x(ph /\ ps))
7		19.8a 1479	. . . . . . 7 |- (ps -> E.yps)
8	7	anim2i 324	. . . . . 6 |- ((ph /\ ps) -> (ph /\ E.yps))
9	8	eximi 1488	. . . . 5 |- (E.x(ph /\ ps) -> E.x(ph /\ E.yps))
10	6, 9	sylbi 114	. . . 4 |- (E.xE.x(ph /\ ps) -> E.x(ph /\ E.yps))
11	4, 10	syl6bir 153	. . 3 |- (A.x x = y -> (E.xE.y(ph /\ ps) -> E.x(ph /\ E.yps)))
12		ax-ial 1424	. . . 4 |- (A.x F/yph -> A.xA.x F/yph)
13		19.40 1519	. . . . . 6 |- (E.y(ph /\ ps) -> (E.yph /\ E.yps))
14		19.9t 1530	. . . . . . . 8 |- ( F/yph -> (E.yph <-> ph))
15	14	biimpd 132	. . . . . . 7 |- ( F/yph -> (E.yph -> ph))
16	15	anim1d 319	. . . . . 6 |- ( F/yph -> ((E.yph /\ E.yps) -> (ph /\ E.yps)))
17	13, 16	syl5 28	. . . . 5 |- ( F/yph -> (E.y(ph /\ ps) -> (ph /\ E.yps)))
18	17	sps 1427	. . . 4 |- (A.x F/yph -> (E.y(ph /\ ps) -> (ph /\ E.yps)))
19	12, 18	eximdh 1499	. . 3 |- (A.x F/yph -> (E.xE.y(ph /\ ps) -> E.x(ph /\ E.yps)))
20	11, 19	jaoi 635	. 2 |- ((A.x x = y \/ A.x F/yph) -> (E.xE.y(ph /\ ps) -> E.x(ph /\ E.yps)))
21	1, 20	ax-mp 7	1 |- (E.xE.y(ph /\ ps) -> E.x(ph /\ E.yps))

Syntax hints:   -> wi 4   /\ wa 97   \/ wo 628  A.wal 1240   F/wnf 1346  E.wex 1378

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424

This theorem depends on definitions:  df-bi 110  df-nf 1347

This theorem is referenced by:  oprabidlem  5479