Theorem eqsbc3rOLD 3460

Description: Obsolete proof of eqsbc3r 3459 as of 7-Jul-2021. This proof was automatically generated from the virtual deduction proof eqsbc3rVD 38097 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
eqsbc3rOLD	⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝑥 ↔ 𝐵 = 𝐴))

Distinct variable group:   𝑥,𝐵

Allowed substitution hints:   𝐴(𝑥)   𝑉(𝑥)

Proof of Theorem eqsbc3rOLD

Step	Hyp	Ref	Expression
1		eqcom 2617	. . . . . 6 ⊢ (𝐵 = 𝑥 ↔ 𝑥 = 𝐵)
2	1	sbcbii 3458	. . . . 5 ⊢ ([𝐴 / 𝑥]𝐵 = 𝑥 ↔ [𝐴 / 𝑥]𝑥 = 𝐵)
3	2	biimpi 205	. . . 4 ⊢ ([𝐴 / 𝑥]𝐵 = 𝑥 → [𝐴 / 𝑥]𝑥 = 𝐵)
4		eqsbc3 3442	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵 ↔ 𝐴 = 𝐵))
5	3, 4	syl5ib 233	. . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝑥 → 𝐴 = 𝐵))
6		eqcom 2617	. . 3 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
7	5, 6	syl6ib 240	. 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝑥 → 𝐵 = 𝐴))
8		idd 24	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐵 = 𝐴 → 𝐵 = 𝐴))
9	8, 6	syl6ibr 241	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐵 = 𝐴 → 𝐴 = 𝐵))
10	9, 4	sylibrd 248	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐵 = 𝐴 → [𝐴 / 𝑥]𝑥 = 𝐵))
11	10, 2	syl6ibr 241	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐵 = 𝐴 → [𝐴 / 𝑥]𝐵 = 𝑥))
12	7, 11	impbid 201	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝑥 ↔ 𝐵 = 𝐴))

Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  [wsbc 3402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175  df-sbc 3403

This theorem is referenced by: (None)