Theorem releqi 5778

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 5777	. 2 ⊢ (𝐴 = 𝐵 → (Rel 𝐴 ↔ Rel 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (Rel 𝐴 ↔ Rel 𝐵)

Syntax hints:   ↔ wb 205   = wceq 1542  Rel wrel 5682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-in 3956  df-ss 3966  df-rel 5684

This theorem is referenced by:  reliun  5817  reluni  5819  relint  5820  reldmmpo  7543  frrlem6  8276  wfrrelOLD  8314  tfrlem6  8382  relsdom  8946  0rest  17375  firest  17378  2oppchomf  17670  oppchofcl  18213  oyoncl  18223  releqg  19055  reldvdsr  20174  restbas  22662  hlimcaui  30489  gonan0  34383  satffunlem2lem2  34397  relbigcup  34869  fnsingle  34891  funimage  34900  colinrel  35029  brcnvrabga  37211  relcoels  37294  iscard4  42284  neicvgnvor  42867  xlimrel  44536