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Theorem dfcleqf 38281
 Description: Equality connective between classes. Same as dfcleq 2604, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypotheses
Ref Expression
dfcleqf.1 𝑥𝐴
dfcleqf.2 𝑥𝐵
Assertion
Ref Expression
dfcleqf (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))

Proof of Theorem dfcleqf
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfcleq 2604 . 2 (𝐴 = 𝐵 ↔ ∀𝑦(𝑦𝐴𝑦𝐵))
2 nfcv 2751 . . . . 5 𝑥𝑦
3 dfcleqf.1 . . . . 5 𝑥𝐴
42, 3nfel 2763 . . . 4 𝑥 𝑦𝐴
5 dfcleqf.2 . . . . 5 𝑥𝐵
62, 5nfel 2763 . . . 4 𝑥 𝑦𝐵
74, 6nfbi 1821 . . 3 𝑥(𝑦𝐴𝑦𝐵)
8 nfv 1830 . . 3 𝑦(𝑥𝐴𝑥𝐵)
9 eleq1 2676 . . . 4 (𝑦 = 𝑥 → (𝑦𝐴𝑥𝐴))
10 eleq1 2676 . . . 4 (𝑦 = 𝑥 → (𝑦𝐵𝑥𝐵))
119, 10bibi12d 334 . . 3 (𝑦 = 𝑥 → ((𝑦𝐴𝑦𝐵) ↔ (𝑥𝐴𝑥𝐵)))
127, 8, 11cbval 2259 . 2 (∀𝑦(𝑦𝐴𝑦𝐵) ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
131, 12bitri 263 1 (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195  ∀wal 1473   = wceq 1475   ∈ wcel 1977  Ⅎwnfc 2738 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-cleq 2603  df-clel 2606  df-nfc 2740 This theorem is referenced by:  ssmapsn  38403  preimagelt  39589  preimalegt  39590  pimrecltpos  39596  pimrecltneg  39610  smfaddlem1  39649
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