Theorem releqi 5359

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 5358	. 2 ⊢ (𝐴 = 𝐵 → (Rel 𝐴 ↔ Rel 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (Rel 𝐴 ↔ Rel 𝐵)

Syntax hints:   ↔ wb 196   = wceq 1632  Rel wrel 5271

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-in 3722  df-ss 3729  df-rel 5273

This theorem is referenced by:  reliun  5395  reluni  5397  relint  5398  reldmmpt2  6937  wfrrel  7590  tfrlem6  7648  relsdom  8130  cda1dif  9210  0rest  16312  firest  16315  2oppchomf  16605  oppchofcl  17121  oyoncl  17131  releqg  17862  reldvdsr  18864  restbas  21184  hlimcaui  28423  frrlem6  32116  relbigcup  32331  fnsingle  32353  funimage  32362  colinrel  32491  cnfinltrel  33570  relcoels  34520  neicvgnvor  38934  xlimrel  40567