Theorem exdistrf 1971

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.)

Hypothesis

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

Assertion

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

Proof of Theorem exdistrf

Step	Hyp	Ref	Expression
1		biidd 228	. . . . 5 |- (A.x x = y -> ((ph /\ ps) <-> (ph /\ ps)))
2	1	drex1 1967	. . . 4 |- (A.x x = y -> (E.x(ph /\ ps) <-> E.y(ph /\ ps)))
3	2	drex2 1968	. . 3 |- (A.x x = y -> (E.xE.x(ph /\ ps) <-> E.xE.y(ph /\ ps)))
4		nfe1 1732	. . . . 5 |- F/xE.x(ph /\ ps)
5	4	19.9 1783	. . . 4 |- (E.xE.x(ph /\ ps) <-> E.x(ph /\ ps))
6		19.8a 1756	. . . . . 6 |- (ps -> E.yps)
7	6	anim2i 552	. . . . 5 |- ((ph /\ ps) -> (ph /\ E.yps))
8	7	eximi 1576	. . . 4 |- (E.x(ph /\ ps) -> E.x(ph /\ E.yps))
9	5, 8	sylbi 187	. . 3 |- (E.xE.x(ph /\ ps) -> E.x(ph /\ E.yps))
10	3, 9	syl6bir 220	. 2 |- (A.x x = y -> (E.xE.y(ph /\ ps) -> E.x(ph /\ E.yps)))
11		nfnae 1956	. . 3 |- F/x -. A.x x = y
12		19.40 1609	. . . 4 |- (E.y(ph /\ ps) -> (E.yph /\ E.yps))
13		exdistrf.1	. . . . . 6 |- (-. A.x x = y -> F/yph)
14	13	19.9d 1782	. . . . 5 |- (-. A.x x = y -> (E.yph -> ph))
15	14	anim1d 547	. . . 4 |- (-. A.x x = y -> ((E.yph /\ E.yps) -> (ph /\ E.yps)))
16	12, 15	syl5 28	. . 3 |- (-. A.x x = y -> (E.y(ph /\ ps) -> (ph /\ E.yps)))
17	11, 16	eximd 1770	. 2 |- (-. A.x x = y -> (E.xE.y(ph /\ ps) -> E.x(ph /\ E.yps)))
18	10, 17	pm2.61i 156	1 |- (E.xE.y(ph /\ ps) -> E.x(ph /\ E.yps))

Syntax hints:  -. wn 3   -> wi 4   /\ wa 358  A.wal 1540  E.wex 1541   F/wnf 1544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  oprabid  5551