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Theorem bj-vtoclgfALT 33005
Description: Alternate proof of vtoclgf 3262. Proof from vtoclgft 3252. (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 471 . 2 (𝑥𝐴 ∧ Ⅎ𝑥𝜓)
4 bj-vtoclgfALT.3 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
54ax-gen 1721 . . 3 𝑥(𝑥 = 𝐴 → (𝜑𝜓))
6 bj-vtoclgfALT.4 . . . 4 𝜑
76ax-gen 1721 . . 3 𝑥𝜑
85, 7pm3.2i 471 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑)
9 vtoclgft 3252 . 2 (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴𝑉) → 𝜓)
103, 8, 9mp3an12 1413 1 (𝐴𝑉𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1480   = wceq 1482  wnf 1707  wcel 1989  wnfc 2750
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-v 3200
This theorem is referenced by: (None)
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