Theorem sbceqalOLD 3779

Description: Obsolete version of sbceqal 3778 as of 26-Oct-2024. (Contributed by Andrew Salmon, 28-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Distinct variable groups:   𝑥,𝐵   𝑥,𝐴

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem sbceqalOLD

Step	Hyp	Ref	Expression
1		spsbc 3724	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵) → [𝐴 / 𝑥](𝑥 = 𝐴 → 𝑥 = 𝐵)))
2		sbcimg 3762	. . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑥 = 𝐴 → 𝑥 = 𝐵) ↔ ([𝐴 / 𝑥]𝑥 = 𝐴 → [𝐴 / 𝑥]𝑥 = 𝐵)))
3		eqid 2738	. . . . 5 ⊢ 𝐴 = 𝐴
4		eqsbc1 3760	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐴 ↔ 𝐴 = 𝐴))
5	3, 4	mpbiri 257	. . . 4 ⊢ (𝐴 ∈ 𝑉 → [𝐴 / 𝑥]𝑥 = 𝐴)
6		pm5.5 361	. . . 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		eqsbc1 3760	. . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵 ↔ 𝐴 = 𝐵))
9	2, 7, 8	3bitrd 304	. 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑥 = 𝐴 → 𝑥 = 𝐵) ↔ 𝐴 = 𝐵))
10	1, 9	sylibd 238	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵) → 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537   = wceq 1539   ∈ wcel 2108  [wsbc 3711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-sbc 3712

This theorem is referenced by: (None)