Theorem dfcleqf 40190

Description: Equality connective between classes. Same as dfcleq 2771, 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 2963	1 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

Syntax hints:   ↔ wb 198  ∀wal 1599   = wceq 1601   ∈ wcel 2107  Ⅎwnfc 2919

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-cleq 2770  df-clel 2774  df-nfc 2921

This theorem is referenced by:  ssmapsn  40333  infnsuprnmpt  40380  preimagelt  41843  preimalegt  41844  pimrecltpos  41850  pimrecltneg  41864  smfaddlem1  41902  smflimsuplem7  41963