Theorem releqi 5202

Description: Equality inference for the relation predicate. (Contributed by NM, 8-Dec-2006.)

Hypothesis

Ref	Expression
releqi.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
releqi	⊢ (Rel 𝐴 ↔ Rel 𝐵)

Proof of Theorem releqi

Step	Hyp	Ref	Expression
1		releqi.1	. 2 ⊢ 𝐴 = 𝐵
2		releq 5201	. 2 ⊢ (𝐴 = 𝐵 → (Rel 𝐴 ↔ Rel 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (Rel 𝐴 ↔ Rel 𝐵)

Syntax hints:   ↔ wb 196   = wceq 1483  Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-in 3581  df-ss 3588  df-rel 5121

This theorem is referenced by:  reliun  5239  reluni  5241  relint  5242  reldmmpt2  6771  wfrrel  7420  tfrlem6  7478  relsdom  7962  cda1dif  8998  0rest  16090  firest  16093  2oppchomf  16384  oppchofcl  16900  oyoncl  16910  releqg  17641  reldvdsr  18644  restbas  20962  hlimcaui  28093  relbigcup  32004  fnsingle  32026  funimage  32035  colinrel  32164  neicvgnvor  38414  xlimrel  40046