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Mirrors > Home > MPE Home > Th. List > Mathboxes > eqvreleqd | Structured version Visualization version GIF version |
Description: Equality theorem for equivalence relation, deduction version. (Contributed by Peter Mazsa, 23-Sep-2021.) |
Ref | Expression |
---|---|
eqvreleqd.1 | ⊢ (𝜑 → 𝑅 = 𝑆) |
Ref | Expression |
---|---|
eqvreleqd | ⊢ (𝜑 → ( EqvRel 𝑅 ↔ EqvRel 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqvreleqd.1 | . 2 ⊢ (𝜑 → 𝑅 = 𝑆) | |
2 | eqvreleq 36701 | . 2 ⊢ (𝑅 = 𝑆 → ( EqvRel 𝑅 ↔ EqvRel 𝑆)) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → ( EqvRel 𝑅 ↔ EqvRel 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1539 EqvRel weqvrel 36336 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2709 ax-sep 5222 ax-nul 5229 ax-pr 5351 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3432 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4258 df-if 4461 df-sn 4563 df-pr 4565 df-op 4569 df-br 5075 df-opab 5137 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-refrel 36616 df-symrel 36644 df-trrel 36674 df-eqvrel 36684 |
This theorem is referenced by: (None) |
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