Theorem eqvreleqd 38132

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

Hypothesis

Ref	Expression
eqvreleqd.1	⊢ (𝜑 → 𝑅 = 𝑆)

Assertion

Ref	Expression
eqvreleqd	⊢ (𝜑 → ( EqvRel 𝑅 ↔ EqvRel 𝑆))

Proof of Theorem eqvreleqd

Step	Hyp	Ref	Expression
1		eqvreleqd.1	. 2 ⊢ (𝜑 → 𝑅 = 𝑆)
2		eqvreleq 38130	. 2 ⊢ (𝑅 = 𝑆 → ( EqvRel 𝑅 ↔ EqvRel 𝑆))
3	1, 2	syl 17	1 ⊢ (𝜑 → ( EqvRel 𝑅 ↔ EqvRel 𝑆))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1533   EqvRel weqvrel 37722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-refrel 38040  df-symrel 38072  df-trrel 38102  df-eqvrel 38113

This theorem is referenced by: (None)