Theorem erALTVeq1i 39385

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 39384	. 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 209   = wceq 1570   ErALTV werALTV 38839

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-ec 8697  df-qs 8701  df-refrel 39222  df-symrel 39254  df-trrel 39288  df-eqvrel 39299  df-dmqs 39353  df-erALTV 39379

This theorem is referenced by:  dfcomember  39387