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Theorem bj-vtoclgfALT 32212
 Description: Alternate proof of vtoclgf 3237. Proof from vtoclgft 3227. (This may have been the original proof before shortening.) (Contributed by BJ, 30-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
bj-vtoclgfALT.1 𝑥𝐴
bj-vtoclgfALT.2 𝑥𝜓
bj-vtoclgfALT.3 (𝑥 = 𝐴 → (𝜑𝜓))
bj-vtoclgfALT.4 𝜑
Assertion
Ref Expression
bj-vtoclgfALT (𝐴𝑉𝜓)

Proof of Theorem bj-vtoclgfALT
StepHypRef Expression
1 bj-vtoclgfALT.1 . . 3 𝑥𝐴
2 bj-vtoclgfALT.2 . . 3 𝑥𝜓
31, 2pm3.2i 470 . 2 (𝑥𝐴 ∧ Ⅎ𝑥𝜓)
4 bj-vtoclgfALT.3 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
54ax-gen 1713 . . 3 𝑥(𝑥 = 𝐴 → (𝜑𝜓))
6 bj-vtoclgfALT.4 . . . 4 𝜑
76ax-gen 1713 . . 3 𝑥𝜑
85, 7pm3.2i 470 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑)
9 vtoclgft 3227 . 2 (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴𝑉) → 𝜓)
103, 8, 9mp3an12 1406 1 (𝐴𝑉𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977  Ⅎwnfc 2738 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175 This theorem is referenced by: (None)
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