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Theorem strcoll2 13211
Description: Version of ax-strcoll 13210 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 2626 . . 3 (𝑧 = 𝑎 → (∀𝑥𝑧𝑦𝜑 ↔ ∀𝑥𝑎𝑦𝜑))
2 rexeq 2627 . . . . . 6 (𝑧 = 𝑎 → (∃𝑥𝑧 𝜑 ↔ ∃𝑥𝑎 𝜑))
32bibi2d 231 . . . . 5 (𝑧 = 𝑎 → ((𝑦𝑏 ↔ ∃𝑥𝑧 𝜑) ↔ (𝑦𝑏 ↔ ∃𝑥𝑎 𝜑)))
43albidv 1796 . . . 4 (𝑧 = 𝑎 → (∀𝑦(𝑦𝑏 ↔ ∃𝑥𝑧 𝜑) ↔ ∀𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑)))
54exbidv 1797 . . 3 (𝑧 = 𝑎 → (∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑧 𝜑) ↔ ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑)))
61, 5imbi12d 233 . 2 (𝑧 = 𝑎 → ((∀𝑥𝑧𝑦𝜑 → ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑧 𝜑)) ↔ (∀𝑥𝑎𝑦𝜑 → ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑))))
7 ax-strcoll 13210 . . 3 𝑧(∀𝑥𝑧𝑦𝜑 → ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑧 𝜑))
87spi 1516 . 2 (∀𝑥𝑧𝑦𝜑 → ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑧 𝜑))
96, 8chvarv 1909 1 (∀𝑥𝑎𝑦𝜑 → ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1329  wex 1468  wral 2416  wrex 2417
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 2121  ax-strcoll 13210
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422
This theorem is referenced by:  strcollnft  13212  strcollnfALT  13214
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