Theorem erALTVeq1d 36762

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

Hypothesis

Ref	Expression
erALTVeq1d.1	⊢ (𝜑 → 𝑅 = 𝑆)

Assertion

Ref	Expression
erALTVeq1d	⊢ (𝜑 → (𝑅 ErALTV 𝐴 ↔ 𝑆 ErALTV 𝐴))

Proof of Theorem erALTVeq1d

Step	Hyp	Ref	Expression
1		erALTVeq1d.1	. 2 ⊢ (𝜑 → 𝑅 = 𝑆)
2		erALTVeq1 36760	. 2 ⊢ (𝑅 = 𝑆 → (𝑅 ErALTV 𝐴 ↔ 𝑆 ErALTV 𝐴))
3	1, 2	syl 17	1 ⊢ (𝜑 → (𝑅 ErALTV 𝐴 ↔ 𝑆 ErALTV 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1541   ErALTV werALTV 36338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-br 5079  df-opab 5141  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-ec 8474  df-qs 8478  df-refrel 36609  df-symrel 36637  df-trrel 36667  df-eqvrel 36677  df-dmqs 36731  df-erALTV 36755

This theorem is referenced by: (None)