Theorem eqsndOLD 4856

Description: Obsolete version of eqsnd 4855 as of 3-Jul-2025. (Contributed by Thierry Arnoux, 10-May-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
eqsnd.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 = 𝐵)
eqsnd.2	⊢ (𝜑 → 𝐵 ∈ 𝐴)

Assertion

Ref	Expression
eqsndOLD	⊢ (𝜑 → 𝐴 = {𝐵})

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥

Proof of Theorem eqsndOLD

Step	Hyp	Ref	Expression
1		eqsnd.1	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 = 𝐵)
2		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵)
3		eqsnd.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
4	3	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝐵 ∈ 𝐴)
5	2, 4	eqeltrd 2844	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝑥 ∈ 𝐴)
6	1, 5	impbida 800	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 = 𝐵))
7		velsn 4664	. . 3 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
8	6, 7	bitr4di 289	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝐵}))
9	8	eqrdv 2738	1 ⊢ (𝜑 → 𝐴 = {𝐵})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {csn 4648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-sn 4649

This theorem is referenced by: (None)