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Theorem ceqsalt 3505
Description: Closed theorem version of ceqsalg 3507. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
Assertion
Ref Expression
ceqsalt ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝑉) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem ceqsalt
StepHypRef Expression
1 biimp 214 . . . . . . 7 ((𝜑𝜓) → (𝜑𝜓))
21imim3i 64 . . . . . 6 ((𝑥 = 𝐴 → (𝜑𝜓)) → ((𝑥 = 𝐴𝜑) → (𝑥 = 𝐴𝜓)))
32al2imi 1818 . . . . 5 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (∀𝑥(𝑥 = 𝐴𝜑) → ∀𝑥(𝑥 = 𝐴𝜓)))
4 elisset 2816 . . . . . 6 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
5 19.23t 2204 . . . . . . 7 (Ⅎ𝑥𝜓 → (∀𝑥(𝑥 = 𝐴𝜓) ↔ (∃𝑥 𝑥 = 𝐴𝜓)))
65biimpd 228 . . . . . 6 (Ⅎ𝑥𝜓 → (∀𝑥(𝑥 = 𝐴𝜓) → (∃𝑥 𝑥 = 𝐴𝜓)))
74, 6syl7 74 . . . . 5 (Ⅎ𝑥𝜓 → (∀𝑥(𝑥 = 𝐴𝜓) → (𝐴𝑉𝜓)))
83, 7sylan9r 510 . . . 4 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → (∀𝑥(𝑥 = 𝐴𝜑) → (𝐴𝑉𝜓)))
98com23 86 . . 3 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) → 𝜓)))
1093impia 1118 . 2 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝑉) → (∀𝑥(𝑥 = 𝐴𝜑) → 𝜓))
11 biimpr 219 . . . . . . . 8 ((𝜑𝜓) → (𝜓𝜑))
1211imim2i 16 . . . . . . 7 ((𝑥 = 𝐴 → (𝜑𝜓)) → (𝑥 = 𝐴 → (𝜓𝜑)))
1312com23 86 . . . . . 6 ((𝑥 = 𝐴 → (𝜑𝜓)) → (𝜓 → (𝑥 = 𝐴𝜑)))
1413alimi 1814 . . . . 5 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → ∀𝑥(𝜓 → (𝑥 = 𝐴𝜑)))
15 19.21t 2200 . . . . 5 (Ⅎ𝑥𝜓 → (∀𝑥(𝜓 → (𝑥 = 𝐴𝜑)) ↔ (𝜓 → ∀𝑥(𝑥 = 𝐴𝜑))))
1614, 15imbitrid 243 . . . 4 (Ⅎ𝑥𝜓 → (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝜓 → ∀𝑥(𝑥 = 𝐴𝜑))))
1716imp 408 . . 3 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → (𝜓 → ∀𝑥(𝑥 = 𝐴𝜑)))
18173adant3 1133 . 2 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝑉) → (𝜓 → ∀𝑥(𝑥 = 𝐴𝜑)))
1910, 18impbid 211 1 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝑉) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088  wal 1540   = wceq 1542  wex 1782  wnf 1786  wcel 2107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-12 2172
This theorem depends on definitions:  df-bi 206  df-an 398  df-3an 1090  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-clel 2811
This theorem is referenced by:  ceqsralt  3506  ceqsalg  3507  vtoclegft  3573
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