Theorem vtoclgOLD 3570

Description: Obsolete version of vtoclg 3553 as of 26-Jan-2025. (Contributed by NM, 17-Apr-1995.) Avoid ax-12 2176. (Revised by SN, 20-Apr-2024.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
vtoclgOLD.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
vtoclgOLD.2	⊢ 𝜑

Assertion

Ref	Expression
vtoclgOLD	⊢ (𝐴 ∈ 𝑉 → 𝜓)

Distinct variable groups:   𝑥,𝐴   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem vtoclgOLD

Step	Hyp	Ref	Expression
1		elisset 2822	. 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
2		vtoclgOLD.2	. . . 4 ⊢ 𝜑
3		vtoclgOLD.1	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4	2, 3	mpbii 233	. . 3 ⊢ (𝑥 = 𝐴 → 𝜓)
5	4	exlimiv 1929	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 → 𝜓)
6	1, 5	syl 17	1 ⊢ (𝐴 ∈ 𝑉 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1539  ∃wex 1778   ∈ wcel 2107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-clel 2815

This theorem is referenced by: (None)