Theorem sbceqal 3800

Description: Class version of one implication of equvelv 2032. (Contributed by Andrew Salmon, 28-Jun-2011.) (Proof shortened by SN, 26-Oct-2024.)

Assertion

Ref	Expression
sbceqal	⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵) → 𝐴 = 𝐵))

Distinct variable groups:   𝑥,𝐵   𝑥,𝐴

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem sbceqal

Step	Hyp	Ref	Expression
1		eqeq1 2738	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐴 ↔ 𝐴 = 𝐴))
2		eqeq1 2738	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐵 ↔ 𝐴 = 𝐵))
3	1, 2	imbi12d 344	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 = 𝐴 → 𝑥 = 𝐵) ↔ (𝐴 = 𝐴 → 𝐴 = 𝐵)))
4		eqid 2734	. . . 4 ⊢ 𝐴 = 𝐴
5	4	a1bi 362	. . 3 ⊢ (𝐴 = 𝐵 ↔ (𝐴 = 𝐴 → 𝐴 = 𝐵))
6	3, 5	bitr4di 289	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 = 𝐴 → 𝑥 = 𝐵) ↔ 𝐴 = 𝐵))
7	6	spcgv 3548	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵) → 𝐴 = 𝐵))

Syntax hints:   → wi 4  ∀wal 1539   = wceq 1541   ∈ wcel 2113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-v 3440

This theorem is referenced by:  sbeqalb  3801