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Theorem bdsep2 10365
Description: Version of ax-bdsep 10363 with one DV condition removed and without initial universal quantifier. Use bdsep1 10364 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 2117 . . . . . 6 (𝑦 = 𝑎 → (𝑥𝑦𝑥𝑎))
21anbi1d 446 . . . . 5 (𝑦 = 𝑎 → ((𝑥𝑦𝜑) ↔ (𝑥𝑎𝜑)))
32bibi2d 225 . . . 4 (𝑦 = 𝑎 → ((𝑥𝑏 ↔ (𝑥𝑦𝜑)) ↔ (𝑥𝑏 ↔ (𝑥𝑎𝜑))))
43albidv 1721 . . 3 (𝑦 = 𝑎 → (∀𝑥(𝑥𝑏 ↔ (𝑥𝑦𝜑)) ↔ ∀𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))))
54exbidv 1722 . 2 (𝑦 = 𝑎 → (∃𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑦𝜑)) ↔ ∃𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))))
6 bdsep2.1 . . 3 BOUNDED 𝜑
76bdsep1 10364 . 2 𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑦𝜑))
85, 7chvarv 1828 1 𝑏𝑥(𝑥𝑏 ↔ (𝑥𝑎𝜑))
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102  wal 1257  wex 1397  BOUNDED wbd 10291
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-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-ext 2038  ax-bdsep 10363
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-cleq 2049  df-clel 2052
This theorem is referenced by:  bdsepnft  10366  bdsepnfALT  10368
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