Theorem eqvreleqi 39396

Description: Equality theorem for equivalence relation, inference version. (Contributed by Peter Mazsa, 23-Sep-2021.)

Hypothesis

Ref	Expression
eqvreleqi.1	⊢ 𝑅 = 𝑆

Assertion

Ref	Expression
eqvreleqi	⊢ ( EqvRel 𝑅 ↔ EqvRel 𝑆)

Proof of Theorem eqvreleqi

Step	Hyp	Ref	Expression
1		eqvreleqi.1	. 2 ⊢ 𝑅 = 𝑆
2		eqvreleq 39395	. 2 ⊢ (𝑅 = 𝑆 → ( EqvRel 𝑅 ↔ EqvRel 𝑆))
3	1, 2	ax-mp 5	1 ⊢ ( EqvRel 𝑅 ↔ EqvRel 𝑆)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   EqvRel weqvrel 38909

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-refrel 39301  df-symrel 39333  df-trrel 39367  df-eqvrel 39378

This theorem is used by:  dfcoeleqvrel  39415  eqvreldmqs2  39470  eldisjim2  39597  eqvrel0  39598  eqvrelid  39601