Theorem extid 39355

Description: Property of identity relation, see also extep 39328, extssr 39628 and the comment of df-ssr 39617. (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 5863	. . . . 5 ⊢ ◡ I = I
2	1	eceq2i 8765	. . . 4 ⊢ [𝐴]◡ I = [𝐴] I
3		ecidsn 8781	. . . 4 ⊢ [𝐴] I = {𝐴}
4	2, 3	eqtri 2783	. . 3 ⊢ [𝐴]◡ I = {𝐴}
5	1	eceq2i 8765	. . . 4 ⊢ [𝐵]◡ I = [𝐵] I
6		ecidsn 8781	. . . 4 ⊢ [𝐵] I = {𝐵}
7	5, 6	eqtri 2783	. . 3 ⊢ [𝐵]◡ I = {𝐵}
8	4, 7	eqeq12i 2778	. 2 ⊢ ([𝐴]◡ I = [𝐵]◡ I ↔ {𝐴} = {𝐵})
9		sneqbg 4797	. 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 1569   ∈ wcel 2144  {csn 4584   I cid 5544  ^◡ccnv 5649  [cec 8720

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ec 8724

This theorem is used by: (None)