Theorem sneqrg 3788

Description: Closed form of sneqr 3786. (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 3629	. . . 4 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
2	1	eqeq1d 2202	. . 3 ⊢ (𝑥 = 𝐴 → ({𝑥} = {𝐵} ↔ {𝐴} = {𝐵}))
3		eqeq1 2200	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐵 ↔ 𝐴 = 𝐵))
4	2, 3	imbi12d 234	. 2 ⊢ (𝑥 = 𝐴 → (({𝑥} = {𝐵} → 𝑥 = 𝐵) ↔ ({𝐴} = {𝐵} → 𝐴 = 𝐵)))
5		vex 2763	. . 3 ⊢ 𝑥 ∈ V
6	5	sneqr 3786	. 2 ⊢ ({𝑥} = {𝐵} → 𝑥 = 𝐵)
7	4, 6	vtoclg 2820	1 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2164  {csn 3618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-sn 3624

This theorem is referenced by:  sneqbg  3789