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Theorem bj-ceqsaltv 35767
Description: Version of bj-ceqsalt 35766 with a disjoint variable condition on 𝑥, 𝑉, removing dependency on df-sb 2069 and df-clab 2711. Prefer its use over bj-ceqsalt 35766 when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-ceqsaltv ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝑉) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑉
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem bj-ceqsaltv
StepHypRef Expression
1 elissetv 2815 . . 3 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
213anim3i 1155 . 2 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝑉) → (Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∃𝑥 𝑥 = 𝐴))
3 bj-ceqsalt0 35764 . 2 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∃𝑥 𝑥 = 𝐴) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
42, 3syl 17 1 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝑉) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1088  wal 1540   = wceq 1542  wex 1782  wnf 1786  wcel 2107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-12 2172
This theorem depends on definitions:  df-bi 206  df-an 398  df-3an 1090  df-ex 1783  df-nf 1787  df-clel 2811
This theorem is referenced by:  bj-ceqsalgvALT  35772
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