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Theorem strcoll2 10495
Description: Version of ax-strcoll 10494 with one DV 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 2522 . . 3 (𝑧 = 𝑎 → (∀𝑥𝑧𝑦𝜑 ↔ ∀𝑥𝑎𝑦𝜑))
2 rexeq 2523 . . . . . 6 (𝑧 = 𝑎 → (∃𝑥𝑧 𝜑 ↔ ∃𝑥𝑎 𝜑))
32bibi2d 225 . . . . 5 (𝑧 = 𝑎 → ((𝑦𝑏 ↔ ∃𝑥𝑧 𝜑) ↔ (𝑦𝑏 ↔ ∃𝑥𝑎 𝜑)))
43albidv 1721 . . . 4 (𝑧 = 𝑎 → (∀𝑦(𝑦𝑏 ↔ ∃𝑥𝑧 𝜑) ↔ ∀𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑)))
54exbidv 1722 . . 3 (𝑧 = 𝑎 → (∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑧 𝜑) ↔ ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑)))
61, 5imbi12d 227 . 2 (𝑧 = 𝑎 → ((∀𝑥𝑧𝑦𝜑 → ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑧 𝜑)) ↔ (∀𝑥𝑎𝑦𝜑 → ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑))))
7 ax-strcoll 10494 . . 3 𝑧(∀𝑥𝑧𝑦𝜑 → ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑧 𝜑))
87spi 1445 . 2 (∀𝑥𝑧𝑦𝜑 → ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑧 𝜑))
96, 8chvarv 1828 1 (∀𝑥𝑎𝑦𝜑 → ∃𝑏𝑦(𝑦𝑏 ↔ ∃𝑥𝑎 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wal 1257  wex 1397  wral 2323  wrex 2324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-strcoll 10494
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329
This theorem is referenced by:  strcollnft  10496  strcollnfALT  10498
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