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Theorem vtoclgOLD 3528
Description: Obsolete version of vtoclg 3527 as of 26-Jan-2025. (Contributed by NM, 17-Apr-1995.) Avoid ax-12 2172. (Revised by SN, 20-Apr-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
vtoclg.1 (𝑥 = 𝐴 → (𝜑𝜓))
vtoclg.2 𝜑
Assertion
Ref Expression
vtoclgOLD (𝐴𝑉𝜓)
Distinct variable groups:   𝑥,𝐴   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem vtoclgOLD
StepHypRef Expression
1 elisset 2816 . 2 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 vtoclg.2 . . . 4 𝜑
3 vtoclg.1 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
42, 3mpbii 232 . . 3 (𝑥 = 𝐴𝜓)
54exlimiv 1934 . 2 (∃𝑥 𝑥 = 𝐴𝜓)
61, 5syl 17 1 (𝐴𝑉𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wex 1782  wcel 2107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-clel 2811
This theorem is referenced by: (None)
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