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Theorem cbvraldva2 3455
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 487 . . . . . . . . 9 ((𝜑𝑥 = 𝑦) → 𝑥 = 𝑦)
2 cbvraldva2.2 . . . . . . . . 9 ((𝜑𝑥 = 𝑦) → 𝐴 = 𝐵)
31, 2eleq12d 2905 . . . . . . . 8 ((𝜑𝑥 = 𝑦) → (𝑥𝐴𝑦𝐵))
4 cbvraldva2.1 . . . . . . . 8 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
53, 4imbi12d 347 . . . . . . 7 ((𝜑𝑥 = 𝑦) → ((𝑥𝐴𝜓) ↔ (𝑦𝐵𝜒)))
65expcom 416 . . . . . 6 (𝑥 = 𝑦 → (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑦𝐵𝜒))))
76pm5.74d 275 . . . . 5 (𝑥 = 𝑦 → ((𝜑 → (𝑥𝐴𝜓)) ↔ (𝜑 → (𝑦𝐵𝜒))))
87cbvalvw 2036 . . . 4 (∀𝑥(𝜑 → (𝑥𝐴𝜓)) ↔ ∀𝑦(𝜑 → (𝑦𝐵𝜒)))
9 19.21v 1933 . . . 4 (∀𝑥(𝜑 → (𝑥𝐴𝜓)) ↔ (𝜑 → ∀𝑥(𝑥𝐴𝜓)))
10 19.21v 1933 . . . 4 (∀𝑦(𝜑 → (𝑦𝐵𝜒)) ↔ (𝜑 → ∀𝑦(𝑦𝐵𝜒)))
118, 9, 103bitr3i 303 . . 3 ((𝜑 → ∀𝑥(𝑥𝐴𝜓)) ↔ (𝜑 → ∀𝑦(𝑦𝐵𝜒)))
1211pm5.74ri 274 . 2 (𝜑 → (∀𝑥(𝑥𝐴𝜓) ↔ ∀𝑦(𝑦𝐵𝜒)))
13 df-ral 3141 . 2 (∀𝑥𝐴 𝜓 ↔ ∀𝑥(𝑥𝐴𝜓))
14 df-ral 3141 . 2 (∀𝑦𝐵 𝜒 ↔ ∀𝑦(𝑦𝐵𝜒))
1512, 13, 143bitr4g 316 1 (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑦𝐵 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wal 1528   = wceq 1530  wcel 2107  wral 3136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1774  df-cleq 2812  df-clel 2891  df-ral 3141
This theorem is referenced by:  cbvraldva  3458  tfrlem3a  8005  mreexexlemd  16907  ismnu  40577
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