Theorem dfcleqf 39255

Description: Equality connective between classes. Same as dfcleq 2616, 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 2790	1 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

Syntax hints:   ↔ wb 196  ∀wal 1481   = wceq 1483   ∈ wcel 1990  Ⅎwnfc 2751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-cleq 2615  df-clel 2618  df-nfc 2753

This theorem is referenced by:  ssmapsn  39408  infnsuprnmpt  39465  preimagelt  40912  preimalegt  40913  pimrecltpos  40919  pimrecltneg  40933  smfaddlem1  40971  smflimsuplem7  41032