Theorem sneqrgOLD 4313

Description: Obsolete proof of sneqrg 4310 as of 23-Jul-2021. (Contributed by Scott Fenton, 1-Apr-2011.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
sneqrgOLD	⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))

Proof of Theorem sneqrgOLD

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 4135	. . . 4 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
2	1	eqeq1d 2612	. . 3 ⊢ (𝑥 = 𝐴 → ({𝑥} = {𝐵} ↔ {𝐴} = {𝐵}))
3		eqeq1 2614	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐵 ↔ 𝐴 = 𝐵))
4	2, 3	imbi12d 333	. 2 ⊢ (𝑥 = 𝐴 → (({𝑥} = {𝐵} → 𝑥 = 𝐵) ↔ ({𝐴} = {𝐵} → 𝐴 = 𝐵)))
5		vex 3176	. . 3 ⊢ 𝑥 ∈ V
6	5	sneqr 4311	. 2 ⊢ ({𝑥} = {𝐵} → 𝑥 = 𝐵)
7	4, 6	vtoclg 3239	1 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  {csn 4125

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-nfc 2740  df-v 3175  df-sn 4126

This theorem is referenced by: (None)