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Theorem bdsep2 13011
Description: Version of ax-bdsep 13009 with one disjoint variable condition removed and without initial universal quantifier. Use bdsep1 13010 when sufficient. (Contributed by BJ, 5-Oct-2019.)
Hypothesis
Ref Expression
bdsep2.1 BOUNDED 𝜑
Assertion
Ref Expression
bdsep2 𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))
Distinct variable groups:   𝑎,𝑏,𝑥   𝜑,𝑏
Allowed substitution hints:   𝜑(𝑥,𝑎)

Proof of Theorem bdsep2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eleq2 2181 . . . . . 6 (𝑦 = 𝑎 → (𝑥𝑦𝑥𝑎))
21anbi1d 460 . . . . 5 (𝑦 = 𝑎 → ((𝑥𝑦𝜑) ↔ (𝑥𝑎𝜑)))
32bibi2d 231 . . . 4 (𝑦 = 𝑎 → ((𝑥𝑏 ↔ (𝑥𝑦𝜑)) ↔ (𝑥𝑏 ↔ (𝑥𝑎𝜑))))
43albidv 1780 . . 3 (𝑦 = 𝑎 → (∀𝑥(𝑥𝑏 ↔ (𝑥𝑦𝜑)) ↔ ∀𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))))
54exbidv 1781 . 2 (𝑦 = 𝑎 → (∃𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑦𝜑)) ↔ ∃𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))))
6 bdsep2.1 . . 3 BOUNDED 𝜑
76bdsep1 13010 . 2 𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑦𝜑))
85, 7chvarv 1889 1 𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wal 1314  wex 1453  BOUNDED wbd 12937
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-5 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-ext 2099  ax-bdsep 13009
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-cleq 2110  df-clel 2113
This theorem is referenced by:  bdsepnft  13012  bdsepnfALT  13014
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