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Theorem cbvraldva2 3170
Description: Rule used to change the bound variable in a restricted universal quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
cbvraldva2.1 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
cbvraldva2.2 ((𝜑𝑥 = 𝑦) → 𝐴 = 𝐵)
Assertion
Ref Expression
cbvraldva2 (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑦𝐵 𝜒))
Distinct variable groups:   𝑦,𝐴   𝜓,𝑦   𝑥,𝐵   𝜒,𝑥   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem cbvraldva2
StepHypRef Expression
1 simpr 477 . . . . 5 ((𝜑𝑥 = 𝑦) → 𝑥 = 𝑦)
2 cbvraldva2.2 . . . . 5 ((𝜑𝑥 = 𝑦) → 𝐴 = 𝐵)
31, 2eleq12d 2693 . . . 4 ((𝜑𝑥 = 𝑦) → (𝑥𝐴𝑦𝐵))
4 cbvraldva2.1 . . . 4 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
53, 4imbi12d 334 . . 3 ((𝜑𝑥 = 𝑦) → ((𝑥𝐴𝜓) ↔ (𝑦𝐵𝜒)))
65cbvaldva 2279 . 2 (𝜑 → (∀𝑥(𝑥𝐴𝜓) ↔ ∀𝑦(𝑦𝐵𝜒)))
7 df-ral 2914 . 2 (∀𝑥𝐴 𝜓 ↔ ∀𝑥(𝑥𝐴𝜓))
8 df-ral 2914 . 2 (∀𝑦𝐵 𝜒 ↔ ∀𝑦(𝑦𝐵𝜒))
96, 7, 83bitr4g 303 1 (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑦𝐵 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1479   = wceq 1481  wcel 1988  wral 2909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1703  df-cleq 2613  df-clel 2616  df-ral 2914
This theorem is referenced by:  cbvraldva  3172  tfrlem3a  7458  mreexexlemd  16285
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