Theorem strcoll2 16370

Description: Version of ax-strcoll 16369 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 2728	. . 3 ⊢ (𝑧 = 𝑎 → (∀𝑥 ∈ 𝑧 ∃𝑦𝜑 ↔ ∀𝑥 ∈ 𝑎 ∃𝑦𝜑))
2		raleq 2728	. . . . 5 ⊢ (𝑧 = 𝑎 → (∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑏 𝜑 ↔ ∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑))
3		rexeq 2729	. . . . . 6 ⊢ (𝑧 = 𝑎 → (∃𝑥 ∈ 𝑧 𝜑 ↔ ∃𝑥 ∈ 𝑎 𝜑))
4	3	ralbidv 2530	. . . . 5 ⊢ (𝑧 = 𝑎 → (∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑧 𝜑 ↔ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑))
5	2, 4	anbi12d 473	. . . 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 1871	. . 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 234	. 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 16369	. . 3 ⊢ ∀𝑧(∀𝑥 ∈ 𝑧 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑧 𝜑))
9	8	spi 1582	. 2 ⊢ (∀𝑥 ∈ 𝑧 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑧 𝜑))
10	7, 9	chvarv 1988	1 ⊢ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑))

Syntax hints:   → wi 4   ∧ wa 104  ∃wex 1538  ∀wral 2508  ∃wrex 2509

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-strcoll 16369

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514

This theorem is referenced by:  strcollnft  16371  strcollnfALT  16373