Theorem strcollnf 13172

Description: Version of ax-strcoll 13169 with one disjoint variable condition removed, the other disjoint variable condition replaced with a non-freeness hypothesis, and without initial universal quantifier. (Contributed by BJ, 21-Oct-2019.)

Hypothesis

Ref	Expression
strcollnf.nf	|- F/ b ph

Assertion

Ref	Expression
strcollnf	|- ( A. x e. a E. y ph -> E. b A. y ( y e. b <-> E. x e. a ph ) )

Distinct variable group:   a, b, x, y

Allowed substitution hints:   ph( x, y, a, b)

Proof of Theorem strcollnf

Step	Hyp	Ref	Expression
1		strcollnft 13171	. 2 |- ( A. x A. y F/ b ph -> ( A. x e. a E. y ph -> E. b A. y ( y e. b <-> E. x e. a ph ) ) )
2		strcollnf.nf	. . 3 |- F/ b ph
3	2	ax-gen 1425	. 2 |- A. y F/ b ph
4	1, 3	mpg 1427	1 |- ( A. x e. a E. y ph -> E. b A. y ( y e. b <-> E. x e. a ph ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1329   F/wnf 1436   E.wex 1468   A.wral 2414   E.wrex 2415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-strcoll 13169

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420

This theorem is referenced by: (None)