Theorem sneqrg 4370

Description: Closed form of sneqr 4371. (Contributed by Scott Fenton, 1-Apr-2011.) (Proof shortened by JJ, 23-Jul-2021.)

Assertion

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

Proof of Theorem sneqrg

Step	Hyp	Ref	Expression
1		snidg 4206	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
2		eleq2 2690	. . 3 ⊢ ({𝐴} = {𝐵} → (𝐴 ∈ {𝐴} ↔ 𝐴 ∈ {𝐵}))
3	1, 2	syl5ibcom 235	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} → 𝐴 ∈ {𝐵}))
4		elsng 4191	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
5	3, 4	sylibd 229	1 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  {csn 4177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sn 4178

This theorem is referenced by:  sneqr  4371  sneqbg  4374  altopth1  32072  altopth2  32073