Theorem erALTVeq1i 38672

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 38671	. 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 206   = wceq 1539   ErALTV werALTV 38209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ec 8748  df-qs 8752  df-refrel 38514  df-symrel 38546  df-trrel 38576  df-eqvrel 38587  df-dmqs 38641  df-erALTV 38666

This theorem is referenced by:  dfcomember  38674