Theorem ideq2 34626

Description: For sets, the identity binary relation is the same as equality. (Contributed by Peter Mazsa, 24-Jun-2020.) (Revised by Peter Mazsa, 18-Dec-2021.)

Assertion

Ref	Expression
ideq2	⊢ (𝐴 ∈ 𝑉 → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))

Proof of Theorem ideq2

Step	Hyp	Ref	Expression
1		brid 34625	. 2 ⊢ (𝐴 I 𝐵 ↔ 𝐵 I 𝐴)
2		ideqg 5505	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐵 I 𝐴 ↔ 𝐵 = 𝐴))
3		eqcom 2831	. . 3 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵)
4	2, 3	syl6bb 279	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐵 I 𝐴 ↔ 𝐴 = 𝐵))
5	1, 4	syl5bb 275	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 198   = wceq 1658   ∈ wcel 2166   class class class wbr 4872   I cid 5248

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pr 5126

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ral 3121  df-rex 3122  df-rab 3125  df-v 3415  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-sn 4397  df-pr 4399  df-op 4403  df-br 4873  df-opab 4935  df-id 5249  df-xp 5347  df-rel 5348  df-cnv 5349

This theorem is referenced by:  br1cossinidres  34746  br1cossxrnidres  34748