Theorem strcollnf 13354

Description: Version of ax-strcoll 13351 with one disjoint variable condition removed, the other disjoint variable condition replaced with a non-freeness hypothesis, and without initial universal quantifier. Version of strcoll2 13352 with the disjoint variable condition on b , ph replaced with a non-freeness hypothesis.

This proof aims to demonstrate a standard technique, but strcoll2 13352 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 13353	. 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 1426	. 2 |- A. y F/ b ph
4	1, 3	mpg 1428	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 103   A.wal 1330   F/wnf 1437   E.wex 1469   A.wral 2417   E.wrex 2418

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-strcoll 13351

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423

This theorem is referenced by: (None)