Theorem sneqrg 3554

Description: Closed form of sneqr 3552. (Contributed by Scott Fenton, 1-Apr-2011.)

Assertion

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

Proof of Theorem sneqrg

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 3409	. . . 4 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
2	1	eqeq1d 2089	. . 3 ⊢ (𝑥 = 𝐴 → ({𝑥} = {𝐵} ↔ {𝐴} = {𝐵}))
3		eqeq1 2087	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐵 ↔ 𝐴 = 𝐵))
4	2, 3	imbi12d 232	. 2 ⊢ (𝑥 = 𝐴 → (({𝑥} = {𝐵} → 𝑥 = 𝐵) ↔ ({𝐴} = {𝐵} → 𝐴 = 𝐵)))
5		vex 2604	. . 3 ⊢ 𝑥 ∈ V
6	5	sneqr 3552	. 2 ⊢ ({𝑥} = {𝐵} → 𝑥 = 𝐵)
7	4, 6	vtoclg 2658	1 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))

Syntax hints:   → wi 4   = wceq 1284   ∈ wcel 1433  {csn 3398

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-sn 3404

This theorem is referenced by:  sneqbg  3555