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Mirrors > Home > MPE Home > Th. List > Mathboxes > erALTVeq1d | Structured version Visualization version GIF version |
Description: Equality theorem for equivalence relation on domain quotient, deduction version. (Contributed by Peter Mazsa, 25-Sep-2021.) |
Ref | Expression |
---|---|
erALTVeq1d.1 | ⊢ (𝜑 → 𝑅 = 𝑆) |
Ref | Expression |
---|---|
erALTVeq1d | ⊢ (𝜑 → (𝑅 ErALTV 𝐴 ↔ 𝑆 ErALTV 𝐴)) |
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)) |
Colors of variables: wff setvar class |
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) |
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