Theorem strcoll2 13569

Description: Version of ax-strcoll 13568 with one disjoint variable condition removed and without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.)

Assertion

Ref	Expression
strcoll2	⊢ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑))

Distinct variable groups:   𝑎,𝑏,𝑥,𝑦   𝜑,𝑏

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

Proof of Theorem strcoll2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		raleq 2652	. . 3 ⊢ (𝑧 = 𝑎 → (∀𝑥 ∈ 𝑧 ∃𝑦𝜑 ↔ ∀𝑥 ∈ 𝑎 ∃𝑦𝜑))
2		raleq 2652	. . . . 5 ⊢ (𝑧 = 𝑎 → (∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑏 𝜑 ↔ ∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑))
3		rexeq 2653	. . . . . 6 ⊢ (𝑧 = 𝑎 → (∃𝑥 ∈ 𝑧 𝜑 ↔ ∃𝑥 ∈ 𝑎 𝜑))
4	3	ralbidv 2457	. . . . 5 ⊢ (𝑧 = 𝑎 → (∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑧 𝜑 ↔ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑))
5	2, 4	anbi12d 465	. . . 4 ⊢ (𝑧 = 𝑎 → ((∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑧 𝜑) ↔ (∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑)))
6	5	exbidv 1805	. . 3 ⊢ (𝑧 = 𝑎 → (∃𝑏(∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑧 𝜑) ↔ ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑)))
7	1, 6	imbi12d 233	. 2 ⊢ (𝑧 = 𝑎 → ((∀𝑥 ∈ 𝑧 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑧 𝜑)) ↔ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑))))
8		ax-strcoll 13568	. . 3 ⊢ ∀𝑧(∀𝑥 ∈ 𝑧 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑧 𝜑))
9	8	spi 1516	. 2 ⊢ (∀𝑥 ∈ 𝑧 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑧 𝜑))
10	7, 9	chvarv 1917	1 ⊢ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑))

Syntax hints:   → wi 4   ∧ wa 103  ∃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 13568

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:  strcollnft  13570  strcollnfALT  13572