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Theorem bdsep2 13255
Description: Version of ax-bdsep 13253 with one disjoint variable condition removed and without initial universal quantifier. Use bdsep1 13254 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 2204 . . . . . 6 (𝑦 = 𝑎 → (𝑥𝑦𝑥𝑎))
21anbi1d 461 . . . . 5 (𝑦 = 𝑎 → ((𝑥𝑦𝜑) ↔ (𝑥𝑎𝜑)))
32bibi2d 231 . . . 4 (𝑦 = 𝑎 → ((𝑥𝑏 ↔ (𝑥𝑦𝜑)) ↔ (𝑥𝑏 ↔ (𝑥𝑎𝜑))))
43albidv 1797 . . 3 (𝑦 = 𝑎 → (∀𝑥(𝑥𝑏 ↔ (𝑥𝑦𝜑)) ↔ ∀𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))))
54exbidv 1798 . 2 (𝑦 = 𝑎 → (∃𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑦𝜑)) ↔ ∃𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))))
6 bdsep2.1 . . 3 BOUNDED 𝜑
76bdsep1 13254 . 2 𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑦𝜑))
85, 7chvarv 1910 1 𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wal 1330  wex 1469  BOUNDED wbd 13181
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 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-ext 2122  ax-bdsep 13253
This theorem depends on definitions:  df-bi 116  df-nf 1438  df-cleq 2133  df-clel 2136
This theorem is referenced by:  bdsepnft  13256  bdsepnfALT  13258
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