Theorem ideqg 2126

Description: For sets, the identity relation is the same as equality.

Assertion

Ref	Expression
ideqg	⊢ ((A ∈ C ∧ B ∈ D) → (AIB ↔ A = B))

Proof of Theorem ideqg

Step	Hyp	Ref	Expression
1		cleq1 1107	. 2 ⊢ (x = A → (x = y ↔ A = y))
2		cleq2 1110	. 2 ⊢ (y = B → (A = y ↔ A = B))
3		df-id 2125	. 2 ⊢ I = {⟨x, y⟩∣x = y}
4	1, 2, 3	brabg 2116	1 ⊢ ((A ∈ C ∧ B ∈ D) → (AIB ↔ A = B))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = weq 797   = wceq 1091   ∈ wcel 1092   class class class wbr 2054  Icid 2057

This theorem is referenced by:  ideq 2127  resieq 2581

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-id 2125

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