Theorem sbceqal 3837

Description: Class version of one implication of equvelv 2038. (Contributed by Andrew Salmon, 28-Jun-2011.)

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		spsbc 3787	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵) → [𝐴 / 𝑥](𝑥 = 𝐴 → 𝑥 = 𝐵)))
2		sbcimg 3822	. . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑥 = 𝐴 → 𝑥 = 𝐵) ↔ ([𝐴 / 𝑥]𝑥 = 𝐴 → [𝐴 / 𝑥]𝑥 = 𝐵)))
3		eqid 2823	. . . . 5 ⊢ 𝐴 = 𝐴
4		eqsbc3 3819	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐴 ↔ 𝐴 = 𝐴))
5	3, 4	mpbiri 260	. . . 4 ⊢ (𝐴 ∈ 𝑉 → [𝐴 / 𝑥]𝑥 = 𝐴)
6		pm5.5 364	. . . 4 ⊢ ([𝐴 / 𝑥]𝑥 = 𝐴 → (([𝐴 / 𝑥]𝑥 = 𝐴 → [𝐴 / 𝑥]𝑥 = 𝐵) ↔ [𝐴 / 𝑥]𝑥 = 𝐵))
7	5, 6	syl 17	. . 3 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]𝑥 = 𝐴 → [𝐴 / 𝑥]𝑥 = 𝐵) ↔ [𝐴 / 𝑥]𝑥 = 𝐵))
8		eqsbc3 3819	. . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵 ↔ 𝐴 = 𝐵))
9	2, 7, 8	3bitrd 307	. 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑥 = 𝐴 → 𝑥 = 𝐵) ↔ 𝐴 = 𝐵))
10	1, 9	sylibd 241	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵) → 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1535   = wceq 1537   ∈ wcel 2114  [wsbc 3774

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-sbc 3775

This theorem is referenced by:  sbeqalb  3838