Theorem extid 39514

Description: Property of identity relation, see also extep 39487, extssr 39787 and the comment of df-ssr 39776. (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 5875	. . . . 5 ⊢ ◡ I = I
2	1	eceq2i 8798	. . . 4 ⊢ [𝐴]◡ I = [𝐴] I
3		ecidsn 8814	. . . 4 ⊢ [𝐴] I = {𝐴}
4	2, 3	eqtri 2786	. . 3 ⊢ [𝐴]◡ I = {𝐴}
5	1	eceq2i 8798	. . . 4 ⊢ [𝐵]◡ I = [𝐵] I
6		ecidsn 8814	. . . 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 4803	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵))
10	8, 9	bitrid 286	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴]◡ I = [𝐵]◡ I ↔ 𝐴 = 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2147  {csn 4588   I cid 5552  ^◡ccnv 5657  [cec 8753

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-ec 8757

This theorem is used by: (None)