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Theorem ceqsralvOLD 3446
Description: Obsolete version of ceqsalv 3443 as of 8-Sep-2024. (Contributed by NM, 21-Jun-2013.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
ceqsralv.2 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
ceqsralvOLD (𝐴𝐵 → (∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ 𝜓))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ceqsralvOLD
StepHypRef Expression
1 nfv 1922 . 2 𝑥𝜓
2 ceqsralv.2 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
32ax-gen 1803 . 2 𝑥(𝑥 = 𝐴 → (𝜑𝜓))
4 ceqsralt 3439 . 2 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ 𝜓))
51, 3, 4mp3an12 1453 1 (𝐴𝐵 → (∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1541   = wceq 1543  wnf 1791  wcel 2110  wral 3061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-12 2175  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1091  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066
This theorem is referenced by: (None)
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