Theorem eqvreleqd 39760

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 39758	. 2 ⊢ (𝑅 = 𝑆 → ( EqvRel 𝑅 ↔ EqvRel 𝑆))
3	1, 2	syl 18	1 ⊢ (𝜑 → ( EqvRel 𝑅 ↔ EqvRel 𝑆))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1569   EqvRel weqvrel 39272

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5251  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-br 5106  df-opab 5170  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-refrel 39664  df-symrel 39696  df-trrel 39730  df-eqvrel 39741

This theorem is used by: (None)