Theorem releqi 5646

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 5645	. 2 ⊢ (𝐴 = 𝐵 → (Rel 𝐴 ↔ Rel 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (Rel 𝐴 ↔ Rel 𝐵)

Syntax hints:   ↔ wb 208   = wceq 1533  Rel wrel 5554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-in 3942  df-ss 3951  df-rel 5556

This theorem is referenced by:  reliun  5683  reluni  5685  relint  5686  reldmmpo  7279  wfrrel  7954  tfrlem6  8012  relsdom  8510  0rest  16697  firest  16700  2oppchomf  16988  oppchofcl  17504  oyoncl  17514  releqg  18321  reldvdsr  19388  restbas  21760  hlimcaui  29007  gonan0  32634  satffunlem2lem2  32648  frrlem6  33123  relbigcup  33353  fnsingle  33375  funimage  33384  colinrel  33513  brcnvrabga  35593  relcoels  35663  iscard4  39893  neicvgnvor  40459  xlimrel  42094