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Theorem strcollnf 13531
 Description: Version of ax-strcoll 13528 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 13529 with the disjoint variable condition on 𝑏, 𝜑 replaced with a non-freeness hypothesis. This proof aims to demonstrate a standard technique, but strcoll2 13529 will generally suffice: since the theorem asserts the existence of a set 𝑏, supposing that that setvar does not occur in the already defined 𝜑 is not a big constraint. (Contributed by BJ, 21-Oct-2019.)
Hypothesis
Ref Expression
strcollnf.nf 𝑏𝜑
Assertion
Ref Expression
strcollnf (∀𝑥𝑎𝑦𝜑 → ∃𝑏(∀𝑥𝑎𝑦𝑏 𝜑 ∧ ∀𝑦𝑏𝑥𝑎 𝜑))
Distinct variable group:   𝑎,𝑏,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑎,𝑏)

Proof of Theorem strcollnf
StepHypRef Expression
1 strcollnft 13530 . 2 (∀𝑥𝑦𝑏𝜑 → (∀𝑥𝑎𝑦𝜑 → ∃𝑏(∀𝑥𝑎𝑦𝑏 𝜑 ∧ ∀𝑦𝑏𝑥𝑎 𝜑)))
2 strcollnf.nf . . 3 𝑏𝜑
32ax-gen 1429 . 2 𝑦𝑏𝜑
41, 3mpg 1431 1 (∀𝑥𝑎𝑦𝜑 → ∃𝑏(∀𝑥𝑎𝑦𝑏 𝜑 ∧ ∀𝑦𝑏𝑥𝑎 𝜑))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1333  Ⅎwnf 1440  ∃wex 1472  ∀wral 2435  ∃wrex 2436 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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139  ax-strcoll 13528 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441 This theorem is referenced by: (None)
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