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Theorem ceqsralt 3260
Description: Restricted quantifier version of ceqsalt 3259. (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 2946 . . . 4 (∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ ∀𝑥(𝑥𝐵 → (𝑥 = 𝐴𝜑)))
2 eleq1 2718 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
32pm5.32ri 671 . . . . . . 7 ((𝑥𝐵𝑥 = 𝐴) ↔ (𝐴𝐵𝑥 = 𝐴))
43imbi1i 338 . . . . . 6 (((𝑥𝐵𝑥 = 𝐴) → 𝜑) ↔ ((𝐴𝐵𝑥 = 𝐴) → 𝜑))
5 impexp 461 . . . . . 6 (((𝑥𝐵𝑥 = 𝐴) → 𝜑) ↔ (𝑥𝐵 → (𝑥 = 𝐴𝜑)))
6 impexp 461 . . . . . 6 (((𝐴𝐵𝑥 = 𝐴) → 𝜑) ↔ (𝐴𝐵 → (𝑥 = 𝐴𝜑)))
74, 5, 63bitr3i 290 . . . . 5 ((𝑥𝐵 → (𝑥 = 𝐴𝜑)) ↔ (𝐴𝐵 → (𝑥 = 𝐴𝜑)))
87albii 1787 . . . 4 (∀𝑥(𝑥𝐵 → (𝑥 = 𝐴𝜑)) ↔ ∀𝑥(𝐴𝐵 → (𝑥 = 𝐴𝜑)))
9 19.21v 1908 . . . 4 (∀𝑥(𝐴𝐵 → (𝑥 = 𝐴𝜑)) ↔ (𝐴𝐵 → ∀𝑥(𝑥 = 𝐴𝜑)))
101, 8, 93bitri 286 . . 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 349 . . 3 (𝐴𝐵 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ (𝐴𝐵 → ∀𝑥(𝑥 = 𝐴𝜑))))
13123ad2ant3 1104 . 2 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ (𝐴𝐵 → ∀𝑥(𝑥 = 𝐴𝜑))))
14 ceqsalt 3259 . 2 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
1511, 13, 143bitr2d 296 1 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054  wal 1521   = wceq 1523  wnf 1748  wcel 2030  wral 2941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-12 2087  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-ral 2946  df-v 3233
This theorem is referenced by:  ceqsralv  3265  cdleme32fva  36042
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