Theorem strcoll2 13170

Description: Version of ax-strcoll 13169 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∀𝑦(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 2624	. . 3 ⊢ (𝑧 = 𝑎 → (∀𝑥 ∈ 𝑧 ∃𝑦𝜑 ↔ ∀𝑥 ∈ 𝑎 ∃𝑦𝜑))
2		rexeq 2625	. . . . . 6 ⊢ (𝑧 = 𝑎 → (∃𝑥 ∈ 𝑧 𝜑 ↔ ∃𝑥 ∈ 𝑎 𝜑))
3	2	bibi2d 231	. . . . 5 ⊢ (𝑧 = 𝑎 → ((𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑧 𝜑) ↔ (𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑎 𝜑)))
4	3	albidv 1796	. . . 4 ⊢ (𝑧 = 𝑎 → (∀𝑦(𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑧 𝜑) ↔ ∀𝑦(𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑎 𝜑)))
5	4	exbidv 1797	. . 3 ⊢ (𝑧 = 𝑎 → (∃𝑏∀𝑦(𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑧 𝜑) ↔ ∃𝑏∀𝑦(𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑎 𝜑)))
6	1, 5	imbi12d 233	. 2 ⊢ (𝑧 = 𝑎 → ((∀𝑥 ∈ 𝑧 ∃𝑦𝜑 → ∃𝑏∀𝑦(𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑧 𝜑)) ↔ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏∀𝑦(𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑎 𝜑))))
7		ax-strcoll 13169	. . 3 ⊢ ∀𝑧(∀𝑥 ∈ 𝑧 ∃𝑦𝜑 → ∃𝑏∀𝑦(𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑧 𝜑))
8	7	spi 1516	. 2 ⊢ (∀𝑥 ∈ 𝑧 ∃𝑦𝜑 → ∃𝑏∀𝑦(𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑧 𝜑))
9	6, 8	chvarv 1907	1 ⊢ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏∀𝑦(𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑎 𝜑))

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1329  ∃wex 1468  ∀wral 2414  ∃wrex 2415

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-strcoll 13169

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420

This theorem is referenced by:  strcollnft  13171  strcollnfALT  13173