Theorem extid 38965

Description: Property of identity relation, see also extep 38938, extssr 39238 and the comment of df-ssr 39227. (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 5871	. . . . 5 ⊢ ◡ I = I
2	1	eceq2i 8733	. . . 4 ⊢ [𝐴]◡ I = [𝐴] I
3		ecidsn 8749	. . . 4 ⊢ [𝐴] I = {𝐴}
4	2, 3	eqtri 2786	. . 3 ⊢ [𝐴]◡ I = {𝐴}
5	1	eceq2i 8733	. . . 4 ⊢ [𝐵]◡ I = [𝐵] I
6		ecidsn 8749	. . . 4 ⊢ [𝐵] I = {𝐵}
7	5, 6	eqtri 2786	. . 3 ⊢ [𝐵]◡ I = {𝐵}
8	4, 7	eqeq12i 2781	. 2 ⊢ ([𝐴]◡ I = [𝐵]◡ I ↔ {𝐴} = {𝐵})
9		sneqbg 4808	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵))
10	8, 9	bitrid 286	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴]◡ I = [𝐵]◡ I ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2143  {csn 4589   I cid 5555  ^◡ccnv 5660  [cec 8688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-ec 8692

This theorem is referenced by: (None)