Theorem sbceqal 3685

Description: Set theory version of sbeqal1 39092. (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 3646	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵) → [𝐴 / 𝑥](𝑥 = 𝐴 → 𝑥 = 𝐵)))
2		sbcimg 3675	. . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑥 = 𝐴 → 𝑥 = 𝐵) ↔ ([𝐴 / 𝑥]𝑥 = 𝐴 → [𝐴 / 𝑥]𝑥 = 𝐵)))
3		eqid 2806	. . . . 5 ⊢ 𝐴 = 𝐴
4		eqsbc3 3673	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐴 ↔ 𝐴 = 𝐴))
5	3, 4	mpbiri 249	. . . 4 ⊢ (𝐴 ∈ 𝑉 → [𝐴 / 𝑥]𝑥 = 𝐴)
6		pm5.5 352	. . . 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 3673	. . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵 ↔ 𝐴 = 𝐵))
9	2, 7, 8	3bitrd 296	. 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑥 = 𝐴 → 𝑥 = 𝐵) ↔ 𝐴 = 𝐵))
10	1, 9	sylibd 230	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵) → 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 197  ∀wal 1635   = wceq 1637   ∈ wcel 2156  [wsbc 3633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-v 3393  df-sbc 3634

This theorem is referenced by:  sbeqalb  3686