Theorem dfcleqf 39776

Description: Equality connective between classes. Same as dfcleq 2765, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Hypotheses

Ref	Expression
dfcleqf.1	⊢ Ⅎ𝑥𝐴
dfcleqf.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
dfcleqf	⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

Proof of Theorem dfcleqf

Step	Hyp	Ref	Expression
1		dfcleqf.1	. 2 ⊢ Ⅎ𝑥𝐴
2		dfcleqf.2	. 2 ⊢ Ⅎ𝑥𝐵
3	1, 2	cleqf 2939	1 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

Syntax hints:   ↔ wb 196  ∀wal 1629   = wceq 1631   ∈ wcel 2145  Ⅎwnfc 2900

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-cleq 2764  df-clel 2767  df-nfc 2902

This theorem is referenced by:  ssmapsn  39925  infnsuprnmpt  39980  preimagelt  41429  preimalegt  41430  pimrecltpos  41436  pimrecltneg  41450  smfaddlem1  41488  smflimsuplem7  41549