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Theorem cleqfOLD 2956
 Description: Obsolete version of cleqf 2955 as of 10-May-2023. (Contributed by NM, 26-May-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Nov-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
cleqf.1 𝑥𝐴
cleqf.2 𝑥𝐵
Assertion
Ref Expression
cleqfOLD (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))

Proof of Theorem cleqfOLD
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cleqf.1 . . 3 𝑥𝐴
21nfcrii 2919 . 2 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
3 cleqf.2 . . 3 𝑥𝐵
43nfcrii 2919 . 2 (𝑦𝐵 → ∀𝑥 𝑦𝐵)
52, 4cleqh 2883 1 (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198  ∀wal 1505   = wceq 1507   ∈ wcel 2050  Ⅎwnfc 2910 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2744 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-cleq 2765  df-clel 2840  df-nfc 2912 This theorem is referenced by: (None)
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