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Theorem cleqh 2850
Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqf 2916. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 14-Nov-2019.) Remove dependency on ax-13 2379. (Revised by BJ, 30-Nov-2020.)
Hypotheses
Ref Expression
cleqh.1 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
cleqh.2 (𝑦𝐵 → ∀𝑥 𝑦𝐵)
Assertion
Ref Expression
cleqh (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem cleqh
StepHypRef Expression
1 dfcleq 2742 . 2 (𝐴 = 𝐵 ↔ ∀𝑦(𝑦𝐴𝑦𝐵))
2 nfv 1980 . . 3 𝑦(𝑥𝐴𝑥𝐵)
3 cleqh.1 . . . . 5 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
43nf5i 2161 . . . 4 𝑥 𝑦𝐴
5 cleqh.2 . . . . 5 (𝑦𝐵 → ∀𝑥 𝑦𝐵)
65nf5i 2161 . . . 4 𝑥 𝑦𝐵
74, 6nfbi 1970 . . 3 𝑥(𝑦𝐴𝑦𝐵)
8 eleq1 2815 . . . 4 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
9 eleq1 2815 . . . 4 (𝑥 = 𝑦 → (𝑥𝐵𝑦𝐵))
108, 9bibi12d 334 . . 3 (𝑥 = 𝑦 → ((𝑥𝐴𝑥𝐵) ↔ (𝑦𝐴𝑦𝐵)))
112, 7, 10cbvalv1 2308 . 2 (∀𝑥(𝑥𝐴𝑥𝐵) ↔ ∀𝑦(𝑦𝐴𝑦𝐵))
121, 11bitr4i 267 1 (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1618   = wceq 1620  wcel 2127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-ext 2728
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1623  df-ex 1842  df-nf 1847  df-cleq 2741  df-clel 2744
This theorem is referenced by:  abeq2  2858  abbi  2863  cleqf  2916  abeq2f  2918  bj-abeq2  33050
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