Theorem cleqf 2939

Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2873. (Contributed by NM, 26-May-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Nov-2019.)

Hypotheses

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

Assertion

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

Proof of Theorem cleqf

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cleqf.1	. . 3 ⊢ Ⅎ𝑥𝐴
2	1	nfcrii 2906	. 2 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)
3		cleqf.2	. . 3 ⊢ Ⅎ𝑥𝐵
4	3	nfcrii 2906	. 2 ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵)
5	2, 4	cleqh 2873	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 835  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:  abid2f  2940  eqvf  3355  eqrd  3771  eq0f  4073  iunab  4700  iinab  4715  mbfposr  23639  mbfinf  23652  itg1climres  23701  bnj1366  31238  bj-rabtrALT  33259  compab  39171  dfcleqf  39776