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Theorem ceqsralt 3499
Description: Restricted quantifier version of ceqsalt 3498. (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 biimt 360 . . . 4 (𝐴𝐵 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ (𝐴𝐵 → ∀𝑥(𝑥 = 𝐴𝜑))))
2 df-ral 3054 . . . . 5 (∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ ∀𝑥(𝑥𝐵 → (𝑥 = 𝐴𝜑)))
3 eleq1 2813 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
43pm5.32ri 575 . . . . . . . 8 ((𝑥𝐵𝑥 = 𝐴) ↔ (𝐴𝐵𝑥 = 𝐴))
54imbi1i 349 . . . . . . 7 (((𝑥𝐵𝑥 = 𝐴) → 𝜑) ↔ ((𝐴𝐵𝑥 = 𝐴) → 𝜑))
6 impexp 450 . . . . . . 7 (((𝑥𝐵𝑥 = 𝐴) → 𝜑) ↔ (𝑥𝐵 → (𝑥 = 𝐴𝜑)))
7 impexp 450 . . . . . . 7 (((𝐴𝐵𝑥 = 𝐴) → 𝜑) ↔ (𝐴𝐵 → (𝑥 = 𝐴𝜑)))
85, 6, 73bitr3i 301 . . . . . 6 ((𝑥𝐵 → (𝑥 = 𝐴𝜑)) ↔ (𝐴𝐵 → (𝑥 = 𝐴𝜑)))
98albii 1813 . . . . 5 (∀𝑥(𝑥𝐵 → (𝑥 = 𝐴𝜑)) ↔ ∀𝑥(𝐴𝐵 → (𝑥 = 𝐴𝜑)))
10 19.21v 1934 . . . . 5 (∀𝑥(𝐴𝐵 → (𝑥 = 𝐴𝜑)) ↔ (𝐴𝐵 → ∀𝑥(𝑥 = 𝐴𝜑)))
112, 9, 103bitrri 298 . . . 4 ((𝐴𝐵 → ∀𝑥(𝑥 = 𝐴𝜑)) ↔ ∀𝑥𝐵 (𝑥 = 𝐴𝜑))
121, 11bitrdi 287 . . 3 (𝐴𝐵 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ ∀𝑥𝐵 (𝑥 = 𝐴𝜑)))
13123ad2ant3 1132 . 2 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ ∀𝑥𝐵 (𝑥 = 𝐴𝜑)))
14 ceqsalt 3498 . 2 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
1513, 14bitr3d 281 1 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1084  wal 1531   = wceq 1533  wnf 1777  wcel 2098  wral 3053
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2163  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1086  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054
This theorem is referenced by:  ceqsralvOLD  3507  cdleme32fva  39812
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