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Theorem ceqsralt 3476
Description: Restricted quantifier version of ceqsalt 3475. (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 3065 . . . 4 (∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ ∀𝑥(𝑥𝐵 → (𝑥 = 𝐴𝜑)))
2 eleq1 2825 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
32pm5.32ri 576 . . . . . . 7 ((𝑥𝐵𝑥 = 𝐴) ↔ (𝐴𝐵𝑥 = 𝐴))
43imbi1i 349 . . . . . 6 (((𝑥𝐵𝑥 = 𝐴) → 𝜑) ↔ ((𝐴𝐵𝑥 = 𝐴) → 𝜑))
5 impexp 451 . . . . . 6 (((𝑥𝐵𝑥 = 𝐴) → 𝜑) ↔ (𝑥𝐵 → (𝑥 = 𝐴𝜑)))
6 impexp 451 . . . . . 6 (((𝐴𝐵𝑥 = 𝐴) → 𝜑) ↔ (𝐴𝐵 → (𝑥 = 𝐴𝜑)))
74, 5, 63bitr3i 300 . . . . 5 ((𝑥𝐵 → (𝑥 = 𝐴𝜑)) ↔ (𝐴𝐵 → (𝑥 = 𝐴𝜑)))
87albii 1821 . . . 4 (∀𝑥(𝑥𝐵 → (𝑥 = 𝐴𝜑)) ↔ ∀𝑥(𝐴𝐵 → (𝑥 = 𝐴𝜑)))
9 19.21v 1942 . . . 4 (∀𝑥(𝐴𝐵 → (𝑥 = 𝐴𝜑)) ↔ (𝐴𝐵 → ∀𝑥(𝑥 = 𝐴𝜑)))
101, 8, 93bitri 296 . . 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 360 . . 3 (𝐴𝐵 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ (𝐴𝐵 → ∀𝑥(𝑥 = 𝐴𝜑))))
13123ad2ant3 1135 . 2 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ (𝐴𝐵 → ∀𝑥(𝑥 = 𝐴𝜑))))
14 ceqsalt 3475 . 2 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
1511, 13, 143bitr2d 306 1 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087  wal 1539   = wceq 1541  wnf 1785  wcel 2106  wral 3064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-12 2171  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1089  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3065
This theorem is referenced by:  ceqsralvOLD  3484  cdleme32fva  38891
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