Theorem eqvreleqi 35832

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 35831	. 2 ⊢ (𝑅 = 𝑆 → ( EqvRel 𝑅 ↔ EqvRel 𝑆))
3	1, 2	ax-mp 5	1 ⊢ ( EqvRel 𝑅 ↔ EqvRel 𝑆)

Syntax hints:   ↔ wb 208   = wceq 1533   EqvRel weqvrel 35464

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  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-br 5059  df-opab 5121  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-refrel 35746  df-symrel 35774  df-trrel 35804  df-eqvrel 35814

This theorem is referenced by:  dfcoeleqvrel  35851