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Theorem cleqh 2904
Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqf 2970. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 14-Nov-2019.) Remove dependency on ax-13 2419. (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 2796 . 2 (𝐴 = 𝐵 ↔ ∀𝑦(𝑦𝐴𝑦𝐵))
2 nfv 2005 . . 3 𝑦(𝑥𝐴𝑥𝐵)
3 cleqh.1 . . . . 5 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
43nf5i 2189 . . . 4 𝑥 𝑦𝐴
5 cleqh.2 . . . . 5 (𝑦𝐵 → ∀𝑥 𝑦𝐵)
65nf5i 2189 . . . 4 𝑥 𝑦𝐵
74, 6nfbi 1995 . . 3 𝑥(𝑦𝐴𝑦𝐵)
8 eleq1w 2864 . . . 4 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
9 eleq1w 2864 . . . 4 (𝑥 = 𝑦 → (𝑥𝐵𝑦𝐵))
108, 9bibi12d 336 . . 3 (𝑥 = 𝑦 → ((𝑥𝐴𝑥𝐵) ↔ (𝑦𝐴𝑦𝐵)))
112, 7, 10cbvalv1 2347 . 2 (∀𝑥(𝑥𝐴𝑥𝐵) ↔ ∀𝑦(𝑦𝐴𝑦𝐵))
121, 11bitr4i 269 1 (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wal 1635   = wceq 1637  wcel 2155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-ext 2781
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-cleq 2795  df-clel 2798
This theorem is referenced by:  abeq2  2912  abbi  2917  cleqf  2970  abeq2f  2972  bj-abeq2  33081
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