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Theorem ceqsralt 2779
Description: Restricted quantifier version of ceqsalt 2778. (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 2473 . . . 4 (∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ ∀𝑥(𝑥𝐵 → (𝑥 = 𝐴𝜑)))
2 eleq1 2252 . . . . . . . . 9 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
32pm5.32ri 455 . . . . . . . 8 ((𝑥𝐵𝑥 = 𝐴) ↔ (𝐴𝐵𝑥 = 𝐴))
43imbi1i 238 . . . . . . 7 (((𝑥𝐵𝑥 = 𝐴) → 𝜑) ↔ ((𝐴𝐵𝑥 = 𝐴) → 𝜑))
5 impexp 263 . . . . . . 7 (((𝑥𝐵𝑥 = 𝐴) → 𝜑) ↔ (𝑥𝐵 → (𝑥 = 𝐴𝜑)))
6 impexp 263 . . . . . . 7 (((𝐴𝐵𝑥 = 𝐴) → 𝜑) ↔ (𝐴𝐵 → (𝑥 = 𝐴𝜑)))
74, 5, 63bitr3i 210 . . . . . 6 ((𝑥𝐵 → (𝑥 = 𝐴𝜑)) ↔ (𝐴𝐵 → (𝑥 = 𝐴𝜑)))
87albii 1481 . . . . 5 (∀𝑥(𝑥𝐵 → (𝑥 = 𝐴𝜑)) ↔ ∀𝑥(𝐴𝐵 → (𝑥 = 𝐴𝜑)))
98a1i 9 . . . 4 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (∀𝑥(𝑥𝐵 → (𝑥 = 𝐴𝜑)) ↔ ∀𝑥(𝐴𝐵 → (𝑥 = 𝐴𝜑))))
101, 9bitrid 192 . . 3 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ ∀𝑥(𝐴𝐵 → (𝑥 = 𝐴𝜑))))
11 19.21v 1884 . . 3 (∀𝑥(𝐴𝐵 → (𝑥 = 𝐴𝜑)) ↔ (𝐴𝐵 → ∀𝑥(𝑥 = 𝐴𝜑)))
1210, 11bitrdi 196 . 2 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ (𝐴𝐵 → ∀𝑥(𝑥 = 𝐴𝜑))))
13 biimt 241 . . 3 (𝐴𝐵 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ (𝐴𝐵 → ∀𝑥(𝑥 = 𝐴𝜑))))
14133ad2ant3 1022 . 2 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ (𝐴𝐵 → ∀𝑥(𝑥 = 𝐴𝜑))))
15 ceqsalt 2778 . 2 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
1612, 14, 153bitr2d 216 1 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 980  wal 1362   = wceq 1364  wnf 1471  wcel 2160  wral 2468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-ral 2473  df-v 2754
This theorem is referenced by:  ceqsralv  2783
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