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Theorem ceqsralt 3444
Description: Restricted quantifier version of ceqsalt 3443. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
Assertion
Ref Expression
ceqsralt ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ 𝜓))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem ceqsralt
StepHypRef Expression
1 df-ral 3088 . . . 4 (∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ ∀𝑥(𝑥𝐵 → (𝑥 = 𝐴𝜑)))
2 eleq1 2848 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
32pm5.32ri 568 . . . . . . 7 ((𝑥𝐵𝑥 = 𝐴) ↔ (𝐴𝐵𝑥 = 𝐴))
43imbi1i 342 . . . . . 6 (((𝑥𝐵𝑥 = 𝐴) → 𝜑) ↔ ((𝐴𝐵𝑥 = 𝐴) → 𝜑))
5 impexp 443 . . . . . 6 (((𝑥𝐵𝑥 = 𝐴) → 𝜑) ↔ (𝑥𝐵 → (𝑥 = 𝐴𝜑)))
6 impexp 443 . . . . . 6 (((𝐴𝐵𝑥 = 𝐴) → 𝜑) ↔ (𝐴𝐵 → (𝑥 = 𝐴𝜑)))
74, 5, 63bitr3i 293 . . . . 5 ((𝑥𝐵 → (𝑥 = 𝐴𝜑)) ↔ (𝐴𝐵 → (𝑥 = 𝐴𝜑)))
87albii 1783 . . . 4 (∀𝑥(𝑥𝐵 → (𝑥 = 𝐴𝜑)) ↔ ∀𝑥(𝐴𝐵 → (𝑥 = 𝐴𝜑)))
9 19.21v 1899 . . . 4 (∀𝑥(𝐴𝐵 → (𝑥 = 𝐴𝜑)) ↔ (𝐴𝐵 → ∀𝑥(𝑥 = 𝐴𝜑)))
101, 8, 93bitri 289 . . 3 (∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ (𝐴𝐵 → ∀𝑥(𝑥 = 𝐴𝜑)))
1110a1i 11 . 2 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ (𝐴𝐵 → ∀𝑥(𝑥 = 𝐴𝜑))))
12 biimt 353 . . 3 (𝐴𝐵 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ (𝐴𝐵 → ∀𝑥(𝑥 = 𝐴𝜑))))
13123ad2ant3 1116 . 2 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ (𝐴𝐵 → ∀𝑥(𝑥 = 𝐴𝜑))))
14 ceqsalt 3443 . 2 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
1511, 13, 143bitr2d 299 1 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  w3a 1069  wal 1506   = wceq 1508  wnf 1747  wcel 2051  wral 3083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-12 2107  ax-ext 2745
This theorem depends on definitions:  df-bi 199  df-an 388  df-3an 1071  df-ex 1744  df-nf 1748  df-cleq 2766  df-clel 2841  df-ral 3088
This theorem is referenced by:  ceqsralv  3449  cdleme32fva  37051
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