Theorem bj-vtoclgfALT 37042

Description: Alternate proof of vtoclgf 3569. Proof from vtoclgft 3552. (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	⊢ Ⅎ𝑥𝐴
bj-vtoclgfALT.2	⊢ Ⅎ𝑥𝜓
bj-vtoclgfALT.3	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
bj-vtoclgfALT.4	⊢ 𝜑

Assertion

Ref	Expression
bj-vtoclgfALT	⊢ (𝐴 ∈ 𝑉 → 𝜓)

Proof of Theorem bj-vtoclgfALT

Step	Hyp	Ref	Expression
1		bj-vtoclgfALT.1	. . 3 ⊢ Ⅎ𝑥𝐴
2		bj-vtoclgfALT.2	. . 3 ⊢ Ⅎ𝑥𝜓
3	1, 2	pm3.2i 470	. 2 ⊢ (Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓)
4		bj-vtoclgfALT.3	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
5	4	ax-gen 1792	. . 3 ⊢ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
6		bj-vtoclgfALT.4	. . . 4 ⊢ 𝜑
7	6	ax-gen 1792	. . 3 ⊢ ∀𝑥𝜑
8	5, 7	pm3.2i 470	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∀𝑥𝜑)
9		vtoclgft 3552	. 2 ⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴 ∈ 𝑉) → 𝜓)
10	3, 8, 9	mp3an12 1450	1 ⊢ (𝐴 ∈ 𝑉 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535   = wceq 1537  Ⅎwnf 1780   ∈ wcel 2106  Ⅎwnfc 2888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-ex 1777  df-nf 1781  df-cleq 2727  df-clel 2814  df-nfc 2890

This theorem is referenced by: (None)