Theorem extid 38292

Description: Property of identity relation, see also extep 38265, extssr 38491 and the comment of df-ssr 38480. (Contributed by Peter Mazsa, 5-Jul-2019.)

Assertion

Ref	Expression
extid	⊢ (𝐴 ∈ 𝑉 → ([𝐴]◡ I = [𝐵]◡ I ↔ 𝐴 = 𝐵))

Proof of Theorem extid

Step	Hyp	Ref	Expression
1		cnvi 6164	. . . . 5 ⊢ ◡ I = I
2	1	eceq2i 8786	. . . 4 ⊢ [𝐴]◡ I = [𝐴] I
3		ecidsn 8799	. . . 4 ⊢ [𝐴] I = {𝐴}
4	2, 3	eqtri 2763	. . 3 ⊢ [𝐴]◡ I = {𝐴}
5	1	eceq2i 8786	. . . 4 ⊢ [𝐵]◡ I = [𝐵] I
6		ecidsn 8799	. . . 4 ⊢ [𝐵] I = {𝐵}
7	5, 6	eqtri 2763	. . 3 ⊢ [𝐵]◡ I = {𝐵}
8	4, 7	eqeq12i 2753	. 2 ⊢ ([𝐴]◡ I = [𝐵]◡ I ↔ {𝐴} = {𝐵})
9		sneqbg 4848	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵))
10	8, 9	bitrid 283	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴]◡ I = [𝐵]◡ I ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537   ∈ wcel 2106  {csn 4631   I cid 5582  ^◡ccnv 5688  [cec 8742

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ec 8746

This theorem is referenced by: (None)