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	⊢ Ⅎ𝑏𝜑

Assertion

Ref	Expression
strcollnf	⊢ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑))

Distinct variable group:   𝑎,𝑏,𝑥,𝑦

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

Proof of Theorem strcollnf

Step	Hyp	Ref	Expression
1		strcollnft 13530	. 2 ⊢ (∀𝑥∀𝑦Ⅎ𝑏𝜑 → (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑)))
2		strcollnf.nf	. . 3 ⊢ Ⅎ𝑏𝜑
3	2	ax-gen 1429	. 2 ⊢ ∀𝑦Ⅎ𝑏𝜑
4	1, 3	mpg 1431	1 ⊢ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑))

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)