Theorem eqvreleqd 38595

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 38593	. 2 ⊢ (𝑅 = 𝑆 → ( EqvRel 𝑅 ↔ EqvRel 𝑆))
3	1, 2	syl 17	1 ⊢ (𝜑 → ( EqvRel 𝑅 ↔ EqvRel 𝑆))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   EqvRel weqvrel 38186

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-refrel 38503  df-symrel 38535  df-trrel 38565  df-eqvrel 38576

This theorem is referenced by: (None)