Theorem erALTVeq1i 37540

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 37539	. 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 205   = wceq 1542   ErALTV werALTV 37069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ec 8705  df-qs 8709  df-refrel 37382  df-symrel 37414  df-trrel 37444  df-eqvrel 37455  df-dmqs 37509  df-erALTV 37534

This theorem is referenced by:  dfcomember  37542