Theorem erALTVeq1i 38778

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 38777	. 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 1541   ErALTV werALTV 38258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5096  df-opab 5158  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-ec 8633  df-qs 8637  df-refrel 38614  df-symrel 38646  df-trrel 38680  df-eqvrel 38691  df-dmqs 38745  df-erALTV 38772

This theorem is referenced by:  dfcomember  38780