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Theorem ceqsalt 2597
Description: Closed theorem version of ceqsalg 2599. (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 elisset 2585 . . . 4 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
213ad2ant3 938 . . 3 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝑉) → ∃𝑥 𝑥 = 𝐴)
3 bi1 115 . . . . . . 7 ((𝜑𝜓) → (𝜑𝜓))
43imim3i 59 . . . . . 6 ((𝑥 = 𝐴 → (𝜑𝜓)) → ((𝑥 = 𝐴𝜑) → (𝑥 = 𝐴𝜓)))
54al2imi 1363 . . . . 5 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (∀𝑥(𝑥 = 𝐴𝜑) → ∀𝑥(𝑥 = 𝐴𝜓)))
653ad2ant2 937 . . . 4 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝑉) → (∀𝑥(𝑥 = 𝐴𝜑) → ∀𝑥(𝑥 = 𝐴𝜓)))
7 19.23t 1583 . . . . 5 (Ⅎ𝑥𝜓 → (∀𝑥(𝑥 = 𝐴𝜓) ↔ (∃𝑥 𝑥 = 𝐴𝜓)))
873ad2ant1 936 . . . 4 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝑉) → (∀𝑥(𝑥 = 𝐴𝜓) ↔ (∃𝑥 𝑥 = 𝐴𝜓)))
96, 8sylibd 142 . . 3 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝑉) → (∀𝑥(𝑥 = 𝐴𝜑) → (∃𝑥 𝑥 = 𝐴𝜓)))
102, 9mpid 41 . 2 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝑉) → (∀𝑥(𝑥 = 𝐴𝜑) → 𝜓))
11 bi2 125 . . . . . . 7 ((𝜑𝜓) → (𝜓𝜑))
1211imim2i 12 . . . . . 6 ((𝑥 = 𝐴 → (𝜑𝜓)) → (𝑥 = 𝐴 → (𝜓𝜑)))
1312com23 76 . . . . 5 ((𝑥 = 𝐴 → (𝜑𝜓)) → (𝜓 → (𝑥 = 𝐴𝜑)))
1413alimi 1360 . . . 4 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → ∀𝑥(𝜓 → (𝑥 = 𝐴𝜑)))
15143ad2ant2 937 . . 3 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝑉) → ∀𝑥(𝜓 → (𝑥 = 𝐴𝜑)))
16 19.21t 1490 . . . 4 (Ⅎ𝑥𝜓 → (∀𝑥(𝜓 → (𝑥 = 𝐴𝜑)) ↔ (𝜓 → ∀𝑥(𝑥 = 𝐴𝜑))))
17163ad2ant1 936 . . 3 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝑉) → (∀𝑥(𝜓 → (𝑥 = 𝐴𝜑)) ↔ (𝜓 → ∀𝑥(𝑥 = 𝐴𝜑))))
1815, 17mpbid 139 . 2 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝑉) → (𝜓 → ∀𝑥(𝑥 = 𝐴𝜑)))
1910, 18impbid 124 1 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝑉) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  w3a 896  wal 1257   = wceq 1259  wnf 1365  wex 1397  wcel 1409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-v 2576
This theorem is referenced by:  ceqsralt  2598
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