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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 (∀𝑥𝑎𝑦𝜑 → ∃𝑏(∀𝑥𝑎𝑦𝑏 𝜑 ∧ ∀𝑦𝑏𝑥𝑎 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.
StepHypRef Expression
1 raleq 2652 . . 3 (𝑧 = 𝑎 → (∀𝑥𝑧𝑦𝜑 ↔ ∀𝑥𝑎𝑦𝜑))
2 raleq 2652 . . . . 5 (𝑧 = 𝑎 → (∀𝑥𝑧𝑦𝑏 𝜑 ↔ ∀𝑥𝑎𝑦𝑏 𝜑))
3 rexeq 2653 . . . . . 6 (𝑧 = 𝑎 → (∃𝑥𝑧 𝜑 ↔ ∃𝑥𝑎 𝜑))
43ralbidv 2457 . . . . 5 (𝑧 = 𝑎 → (∀𝑦𝑏𝑥𝑧 𝜑 ↔ ∀𝑦𝑏𝑥𝑎 𝜑))
52, 4anbi12d 465 . . . 4 (𝑧 = 𝑎 → ((∀𝑥𝑧𝑦𝑏 𝜑 ∧ ∀𝑦𝑏𝑥𝑧 𝜑) ↔ (∀𝑥𝑎𝑦𝑏 𝜑 ∧ ∀𝑦𝑏𝑥𝑎 𝜑)))
65exbidv 1805 . . 3 (𝑧 = 𝑎 → (∃𝑏(∀𝑥𝑧𝑦𝑏 𝜑 ∧ ∀𝑦𝑏𝑥𝑧 𝜑) ↔ ∃𝑏(∀𝑥𝑎𝑦𝑏 𝜑 ∧ ∀𝑦𝑏𝑥𝑎 𝜑)))
71, 6imbi12d 233 . 2 (𝑧 = 𝑎 → ((∀𝑥𝑧𝑦𝜑 → ∃𝑏(∀𝑥𝑧𝑦𝑏 𝜑 ∧ ∀𝑦𝑏𝑥𝑧 𝜑)) ↔ (∀𝑥𝑎𝑦𝜑 → ∃𝑏(∀𝑥𝑎𝑦𝑏 𝜑 ∧ ∀𝑦𝑏𝑥𝑎 𝜑))))
8 ax-strcoll 13568 . . 3 𝑧(∀𝑥𝑧𝑦𝜑 → ∃𝑏(∀𝑥𝑧𝑦𝑏 𝜑 ∧ ∀𝑦𝑏𝑥𝑧 𝜑))
98spi 1516 . 2 (∀𝑥𝑧𝑦𝜑 → ∃𝑏(∀𝑥𝑧𝑦𝑏 𝜑 ∧ ∀𝑦𝑏𝑥𝑧 𝜑))
107, 9chvarv 1917 1 (∀𝑥𝑎𝑦𝜑 → ∃𝑏(∀𝑥𝑎𝑦𝑏 𝜑 ∧ ∀𝑦𝑏𝑥𝑎 𝜑))
Colors of variables: wff set class
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
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