Theorem erALTVeq1i 39215

Description: Equality theorem for equivalence relation on domain quotient, inference version. (Contributed by Peter Mazsa, 25-Sep-2021.)

Hypothesis

Ref	Expression
erALTVeq1i.1	⊢ 𝑅 = 𝑆

Assertion

Ref	Expression
erALTVeq1i	⊢ (𝑅 ErALTV 𝐴 ↔ 𝑆 ErALTV 𝐴)

Proof of Theorem erALTVeq1i

Step	Hyp	Ref	Expression
1		erALTVeq1i.1	. 2 ⊢ 𝑅 = 𝑆
2		erALTVeq1 39214	. 2 ⊢ (𝑅 = 𝑆 → (𝑅 ErALTV 𝐴 ↔ 𝑆 ErALTV 𝐴))
3	1, 2	ax-mp 5	1 ⊢ (𝑅 ErALTV 𝐴 ↔ 𝑆 ErALTV 𝐴)

Syntax hints:   ↔ wb 208   = wceq 1559   ErALTV werALTV 38669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5243  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-ec 8674  df-qs 8678  df-refrel 39052  df-symrel 39084  df-trrel 39118  df-eqvrel 39129  df-dmqs 39183  df-erALTV 39209

This theorem is referenced by:  dfcomember  39217