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Theorem vtoclgftOLD 3533
Description: Obsolete version of vtoclgft 3532. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by JJ, 11-Aug-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
vtoclgftOLD (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴𝑉) → 𝜓)

Proof of Theorem vtoclgftOLD
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 elisset 3484 . . . . 5 (𝐴𝑉 → ∃𝑧 𝑧 = 𝐴)
2 nfnfc1 2976 . . . . . 6 𝑥𝑥𝐴
3 nfcvd 2974 . . . . . . 7 (𝑥𝐴𝑥𝑧)
4 id 22 . . . . . . 7 (𝑥𝐴𝑥𝐴)
53, 4nfeqd 2983 . . . . . 6 (𝑥𝐴 → Ⅎ𝑥 𝑧 = 𝐴)
6 eqeq1 2824 . . . . . . 7 (𝑧 = 𝑥 → (𝑧 = 𝐴𝑥 = 𝐴))
76a1i 11 . . . . . 6 (𝑥𝐴 → (𝑧 = 𝑥 → (𝑧 = 𝐴𝑥 = 𝐴)))
82, 5, 7cbvexd 2429 . . . . 5 (𝑥𝐴 → (∃𝑧 𝑧 = 𝐴 ↔ ∃𝑥 𝑥 = 𝐴))
91, 8syl5ib 246 . . . 4 (𝑥𝐴 → (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴))
109ad2antrr 724 . . 3 (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑)) → (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴))
11103impia 1113 . 2 (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴𝑉) → ∃𝑥 𝑥 = 𝐴)
12 biimp 217 . . . . . . . . 9 ((𝜑𝜓) → (𝜑𝜓))
1312imim2i 16 . . . . . . . 8 ((𝑥 = 𝐴 → (𝜑𝜓)) → (𝑥 = 𝐴 → (𝜑𝜓)))
1413com23 86 . . . . . . 7 ((𝑥 = 𝐴 → (𝜑𝜓)) → (𝜑 → (𝑥 = 𝐴𝜓)))
1514imp 409 . . . . . 6 (((𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝜑) → (𝑥 = 𝐴𝜓))
1615alanimi 1817 . . . . 5 ((∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) → ∀𝑥(𝑥 = 𝐴𝜓))
17 19.23t 2210 . . . . . 6 (Ⅎ𝑥𝜓 → (∀𝑥(𝑥 = 𝐴𝜓) ↔ (∃𝑥 𝑥 = 𝐴𝜓)))
1817adantl 484 . . . . 5 ((𝑥𝐴 ∧ Ⅎ𝑥𝜓) → (∀𝑥(𝑥 = 𝐴𝜓) ↔ (∃𝑥 𝑥 = 𝐴𝜓)))
1916, 18syl5ib 246 . . . 4 ((𝑥𝐴 ∧ Ⅎ𝑥𝜓) → ((∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) → (∃𝑥 𝑥 = 𝐴𝜓)))
2019imp 409 . . 3 (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑)) → (∃𝑥 𝑥 = 𝐴𝜓))
21203adant3 1128 . 2 (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴𝑉) → (∃𝑥 𝑥 = 𝐴𝜓))
2211, 21mpd 15 1 (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴𝑉) → 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1083  wal 1535   = wceq 1537  wex 1780  wnf 1784  wcel 2114  wnfc 2957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-13 2390  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-ex 1781  df-nf 1785  df-cleq 2813  df-clel 2891  df-nfc 2959
This theorem is referenced by: (None)
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