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Theorem vtoclgft 3553
Description: Closed theorem form of vtoclgf 3565. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by JJ, 11-Aug-2021.) Avoid ax-13 2386. (Revised by Gino Giotto, 6-Oct-2023.)
Assertion
Ref Expression
vtoclgft (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴𝑉) → 𝜓)

Proof of Theorem vtoclgft
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 elisset 3505 . . . . 5 (𝐴𝑉 → ∃𝑧 𝑧 = 𝐴)
2 nfv 1911 . . . . . . . . 9 𝑧𝑥𝐴
3 nfnfc1 2980 . . . . . . . . 9 𝑥𝑥𝐴
4 nfcvd 2978 . . . . . . . . . . 11 (𝑥𝐴𝑥𝑧)
5 id 22 . . . . . . . . . . 11 (𝑥𝐴𝑥𝐴)
64, 5nfeqd 2988 . . . . . . . . . 10 (𝑥𝐴 → Ⅎ𝑥 𝑧 = 𝐴)
76nfnd 1854 . . . . . . . . 9 (𝑥𝐴 → Ⅎ𝑥 ¬ 𝑧 = 𝐴)
8 nfvd 1912 . . . . . . . . 9 (𝑥𝐴 → Ⅎ𝑧 ¬ 𝑥 = 𝐴)
9 eqeq1 2825 . . . . . . . . . . . 12 (𝑧 = 𝑥 → (𝑧 = 𝐴𝑥 = 𝐴))
109a1i 11 . . . . . . . . . . 11 (𝑥𝐴 → (𝑧 = 𝑥 → (𝑧 = 𝐴𝑥 = 𝐴)))
11 notbi 321 . . . . . . . . . . 11 ((𝑧 = 𝐴𝑥 = 𝐴) ↔ (¬ 𝑧 = 𝐴 ↔ ¬ 𝑥 = 𝐴))
1210, 11syl6ib 253 . . . . . . . . . 10 (𝑥𝐴 → (𝑧 = 𝑥 → (¬ 𝑧 = 𝐴 ↔ ¬ 𝑥 = 𝐴)))
13 biimp 217 . . . . . . . . . 10 ((¬ 𝑧 = 𝐴 ↔ ¬ 𝑥 = 𝐴) → (¬ 𝑧 = 𝐴 → ¬ 𝑥 = 𝐴))
1412, 13syl6 35 . . . . . . . . 9 (𝑥𝐴 → (𝑧 = 𝑥 → (¬ 𝑧 = 𝐴 → ¬ 𝑥 = 𝐴)))
152, 3, 7, 8, 14cbv1v 2352 . . . . . . . 8 (𝑥𝐴 → (∀𝑧 ¬ 𝑧 = 𝐴 → ∀𝑥 ¬ 𝑥 = 𝐴))
16 equcomi 2020 . . . . . . . . . 10 (𝑥 = 𝑧𝑧 = 𝑥)
17 biimpr 222 . . . . . . . . . 10 ((¬ 𝑧 = 𝐴 ↔ ¬ 𝑥 = 𝐴) → (¬ 𝑥 = 𝐴 → ¬ 𝑧 = 𝐴))
1816, 12, 17syl56 36 . . . . . . . . 9 (𝑥𝐴 → (𝑥 = 𝑧 → (¬ 𝑥 = 𝐴 → ¬ 𝑧 = 𝐴)))
193, 2, 8, 7, 18cbv1v 2352 . . . . . . . 8 (𝑥𝐴 → (∀𝑥 ¬ 𝑥 = 𝐴 → ∀𝑧 ¬ 𝑧 = 𝐴))
2015, 19impbid 214 . . . . . . 7 (𝑥𝐴 → (∀𝑧 ¬ 𝑧 = 𝐴 ↔ ∀𝑥 ¬ 𝑥 = 𝐴))
21 alnex 1778 . . . . . . 7 (∀𝑧 ¬ 𝑧 = 𝐴 ↔ ¬ ∃𝑧 𝑧 = 𝐴)
22 alnex 1778 . . . . . . 7 (∀𝑥 ¬ 𝑥 = 𝐴 ↔ ¬ ∃𝑥 𝑥 = 𝐴)
2320, 21, 223bitr3g 315 . . . . . 6 (𝑥𝐴 → (¬ ∃𝑧 𝑧 = 𝐴 ↔ ¬ ∃𝑥 𝑥 = 𝐴))
2423con4bid 319 . . . . 5 (𝑥𝐴 → (∃𝑧 𝑧 = 𝐴 ↔ ∃𝑥 𝑥 = 𝐴))
251, 24syl5ib 246 . . . 4 (𝑥𝐴 → (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴))
2625ad2antrr 724 . . 3 (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑)) → (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴))
27263impia 1113 . 2 (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴𝑉) → ∃𝑥 𝑥 = 𝐴)
28 biimp 217 . . . . . . . . 9 ((𝜑𝜓) → (𝜑𝜓))
2928imim2i 16 . . . . . . . 8 ((𝑥 = 𝐴 → (𝜑𝜓)) → (𝑥 = 𝐴 → (𝜑𝜓)))
3029com23 86 . . . . . . 7 ((𝑥 = 𝐴 → (𝜑𝜓)) → (𝜑 → (𝑥 = 𝐴𝜓)))
3130imp 409 . . . . . 6 (((𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝜑) → (𝑥 = 𝐴𝜓))
3231alanimi 1813 . . . . 5 ((∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) → ∀𝑥(𝑥 = 𝐴𝜓))
33 19.23t 2206 . . . . . 6 (Ⅎ𝑥𝜓 → (∀𝑥(𝑥 = 𝐴𝜓) ↔ (∃𝑥 𝑥 = 𝐴𝜓)))
3433adantl 484 . . . . 5 ((𝑥𝐴 ∧ Ⅎ𝑥𝜓) → (∀𝑥(𝑥 = 𝐴𝜓) ↔ (∃𝑥 𝑥 = 𝐴𝜓)))
3532, 34syl5ib 246 . . . 4 ((𝑥𝐴 ∧ Ⅎ𝑥𝜓) → ((∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) → (∃𝑥 𝑥 = 𝐴𝜓)))
3635imp 409 . . 3 (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑)) → (∃𝑥 𝑥 = 𝐴𝜓))
37363adant3 1128 . 2 (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴𝑉) → (∃𝑥 𝑥 = 𝐴𝜓))
3827, 37mpd 15 1 (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴𝑉) → 𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  w3a 1083  wal 1531   = wceq 1533  wex 1776  wnf 1780  wcel 2110  wnfc 2961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-ex 1777  df-nf 1781  df-cleq 2814  df-clel 2893  df-nfc 2963
This theorem is referenced by:  vtocldf  3555  bj-vtoclgfALT  34351
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