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Theorem cleqh 2153
Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqf 2217. (Contributed by NM, 5-Aug-1993.)
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 2050 . 2 (𝐴 = 𝐵 ↔ ∀𝑦(𝑦𝐴𝑦𝐵))
2 ax-17 1435 . . . 4 ((𝑥𝐴𝑥𝐵) → ∀𝑦(𝑥𝐴𝑥𝐵))
3 dfbi2 374 . . . . 5 ((𝑦𝐴𝑦𝐵) ↔ ((𝑦𝐴𝑦𝐵) ∧ (𝑦𝐵𝑦𝐴)))
4 cleqh.1 . . . . . . 7 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
5 cleqh.2 . . . . . . 7 (𝑦𝐵 → ∀𝑥 𝑦𝐵)
64, 5hbim 1453 . . . . . 6 ((𝑦𝐴𝑦𝐵) → ∀𝑥(𝑦𝐴𝑦𝐵))
75, 4hbim 1453 . . . . . 6 ((𝑦𝐵𝑦𝐴) → ∀𝑥(𝑦𝐵𝑦𝐴))
86, 7hban 1455 . . . . 5 (((𝑦𝐴𝑦𝐵) ∧ (𝑦𝐵𝑦𝐴)) → ∀𝑥((𝑦𝐴𝑦𝐵) ∧ (𝑦𝐵𝑦𝐴)))
93, 8hbxfrbi 1377 . . . 4 ((𝑦𝐴𝑦𝐵) → ∀𝑥(𝑦𝐴𝑦𝐵))
10 eleq1 2116 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
11 eleq1 2116 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝐵𝑦𝐵))
1210, 11bibi12d 228 . . . . 5 (𝑥 = 𝑦 → ((𝑥𝐴𝑥𝐵) ↔ (𝑦𝐴𝑦𝐵)))
1312biimpd 136 . . . 4 (𝑥 = 𝑦 → ((𝑥𝐴𝑥𝐵) → (𝑦𝐴𝑦𝐵)))
142, 9, 13cbv3h 1647 . . 3 (∀𝑥(𝑥𝐴𝑥𝐵) → ∀𝑦(𝑦𝐴𝑦𝐵))
1512equcoms 1610 . . . . 5 (𝑦 = 𝑥 → ((𝑥𝐴𝑥𝐵) ↔ (𝑦𝐴𝑦𝐵)))
1615biimprd 151 . . . 4 (𝑦 = 𝑥 → ((𝑦𝐴𝑦𝐵) → (𝑥𝐴𝑥𝐵)))
179, 2, 16cbv3h 1647 . . 3 (∀𝑦(𝑦𝐴𝑦𝐵) → ∀𝑥(𝑥𝐴𝑥𝐵))
1814, 17impbii 121 . 2 (∀𝑥(𝑥𝐴𝑥𝐵) ↔ ∀𝑦(𝑦𝐴𝑦𝐵))
191, 18bitr4i 180 1 (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wal 1257   = wceq 1259  wcel 1409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-cleq 2049  df-clel 2052
This theorem is referenced by:  abeq2  2162
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