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Theorem strcoll2 11524
Description: Version of ax-strcoll 11523 with one disjoint variable condition removed and without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.)
Assertion
Ref Expression
strcoll2 (∀𝑥𝑎𝑦𝜑 → ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 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 2562 . . 3 (𝑧 = 𝑎 → (∀𝑥𝑧𝑦𝜑 ↔ ∀𝑥𝑎𝑦𝜑))
2 rexeq 2563 . . . . . 6 (𝑧 = 𝑎 → (∃𝑥𝑧 𝜑 ↔ ∃𝑥𝑎 𝜑))
32bibi2d 230 . . . . 5 (𝑧 = 𝑎 → ((𝑦𝑏 ↔ ∃𝑥𝑧 𝜑) ↔ (𝑦𝑏 ↔ ∃𝑥𝑎 𝜑)))
43albidv 1752 . . . 4 (𝑧 = 𝑎 → (∀𝑦(𝑦𝑏 ↔ ∃𝑥𝑧 𝜑) ↔ ∀𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑)))
54exbidv 1753 . . 3 (𝑧 = 𝑎 → (∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑧 𝜑) ↔ ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑)))
61, 5imbi12d 232 . 2 (𝑧 = 𝑎 → ((∀𝑥𝑧𝑦𝜑 → ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑧 𝜑)) ↔ (∀𝑥𝑎𝑦𝜑 → ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑))))
7 ax-strcoll 11523 . . 3 𝑧(∀𝑥𝑧𝑦𝜑 → ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑧 𝜑))
87spi 1474 . 2 (∀𝑥𝑧𝑦𝜑 → ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑧 𝜑))
96, 8chvarv 1860 1 (∀𝑥𝑎𝑦𝜑 → ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1287  wex 1426  wral 2359  wrex 2360
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-strcoll 11523
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365
This theorem is referenced by:  strcollnft  11525  strcollnfALT  11527
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