Theorem sneqrg 3697

Description: Closed form of sneqr 3695. (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 3543	. . . 4 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
2	1	eqeq1d 2149	. . 3 ⊢ (𝑥 = 𝐴 → ({𝑥} = {𝐵} ↔ {𝐴} = {𝐵}))
3		eqeq1 2147	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐵 ↔ 𝐴 = 𝐵))
4	2, 3	imbi12d 233	. 2 ⊢ (𝑥 = 𝐴 → (({𝑥} = {𝐵} → 𝑥 = 𝐵) ↔ ({𝐴} = {𝐵} → 𝐴 = 𝐵)))
5		vex 2692	. . 3 ⊢ 𝑥 ∈ V
6	5	sneqr 3695	. 2 ⊢ ({𝑥} = {𝐵} → 𝑥 = 𝐵)
7	4, 6	vtoclg 2749	1 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))

Syntax hints:   → wi 4   = wceq 1332   ∈ wcel 1481  {csn 3532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-sn 3538

This theorem is referenced by:  sneqbg  3698