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Theorem cbv2h 1650
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)
Hypotheses
Ref Expression
cbv2h.1 (𝜑 → (𝜓 → ∀𝑦𝜓))
cbv2h.2 (𝜑 → (𝜒 → ∀𝑥𝜒))
cbv2h.3 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
Assertion
Ref Expression
cbv2h (∀𝑥𝑦𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Proof of Theorem cbv2h
StepHypRef Expression
1 cbv2h.1 . . 3 (𝜑 → (𝜓 → ∀𝑦𝜓))
2 cbv2h.2 . . 3 (𝜑 → (𝜒 → ∀𝑥𝜒))
3 cbv2h.3 . . . 4 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
4 bi1 115 . . . 4 ((𝜓𝜒) → (𝜓𝜒))
53, 4syl6 33 . . 3 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
61, 2, 5cbv1h 1649 . 2 (∀𝑥𝑦𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))
7 equcomi 1608 . . . . 5 (𝑦 = 𝑥𝑥 = 𝑦)
8 bi2 125 . . . . 5 ((𝜓𝜒) → (𝜒𝜓))
97, 3, 8syl56 34 . . . 4 (𝜑 → (𝑦 = 𝑥 → (𝜒𝜓)))
102, 1, 9cbv1h 1649 . . 3 (∀𝑦𝑥𝜑 → (∀𝑦𝜒 → ∀𝑥𝜓))
1110a7s 1359 . 2 (∀𝑥𝑦𝜑 → (∀𝑦𝜒 → ∀𝑥𝜓))
126, 11impbid 124 1 (∀𝑥𝑦𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wal 1257
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
This theorem depends on definitions:  df-bi 114  df-nf 1366
This theorem is referenced by:  cbv2  1651
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