Theorem strcollnf 15008

Description: Version of ax-strcoll 15005 with one disjoint variable condition removed, the other disjoint variable condition replaced with a nonfreeness hypothesis, and without initial universal quantifier. Version of strcoll2 15006 with the disjoint variable condition on b , ph replaced with a nonfreeness hypothesis.

This proof aims to demonstrate a standard technique, but strcoll2 15006 will generally suffice: since the theorem asserts the existence of a set b, supposing that that setvar does not occur in the already defined ph is not a big constraint. (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. x e. a E. y e. b ph /\ A. 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 15007	. 2 |- ( A. x A. y F/ b ph -> ( A. x e. a E. y ph -> E. b ( A. x e. a E. y e. b ph /\ A. y e. b E. x e. a ph ) ) )
2		strcollnf.nf	. . 3 |- F/ b ph
3	2	ax-gen 1459	. 2 |- A. y F/ b ph
4	1, 3	mpg 1461	1 |- ( A. x e. a E. y ph -> E. b ( A. x e. a E. y e. b ph /\ A. y e. b E. x e. a ph ) )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1361   F/wnf 1470   E.wex 1502   A.wral 2465   E.wrex 2466

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169  ax-strcoll 15005

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471

This theorem is referenced by: (None)