Theorem eqvreleqi 39008

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 39007	. 2 ⊢ (𝑅 = 𝑆 → ( EqvRel 𝑅 ↔ EqvRel 𝑆))
3	1, 2	ax-mp 5	1 ⊢ ( EqvRel 𝑅 ↔ EqvRel 𝑆)

Syntax hints:   ↔ wb 206   = wceq 1542   EqvRel weqvrel 38521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-refrel 38913  df-symrel 38945  df-trrel 38979  df-eqvrel 38990

This theorem is referenced by:  dfcoeleqvrel  39027  eqvreldmqs2  39082  eldisjim2  39209  eqvrel0  39210  eqvrelid  39213