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Mirrors > Home > MPE Home > Th. List > eceq1d | Structured version Visualization version GIF version |
Description: Equality theorem for equivalence class (deduction form). (Contributed by Jim Kingdon, 31-Dec-2019.) |
Ref | Expression |
---|---|
eceq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
eceq1d | ⊢ (𝜑 → [𝐴]𝐶 = [𝐵]𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eceq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | eceq1 7669 | . 2 ⊢ (𝐴 = 𝐵 → [𝐴]𝐶 = [𝐵]𝐶) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → [𝐴]𝐶 = [𝐵]𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 [cec 7627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ec 7631 |
This theorem is referenced by: brecop 7727 eroveu 7729 erov 7731 ecovcom 7741 ecovass 7742 ecovdi 7743 addsrmo 9773 mulsrmo 9774 addsrpr 9775 mulsrpr 9776 supsrlem 9811 supsr 9812 qus0 17475 qusinv 17476 qussub 17477 sylow2blem2 17859 frgpadd 17999 vrgpval 18003 vrgpinv 18005 frgpup3lem 18013 qusabl 18091 quscrng 19061 qustgplem 21734 pi1addval 22656 pi1xfrf 22661 pi1xfrval 22662 pi1xfrcnvlem 22664 pi1xfrcnv 22665 pi1cof 22667 pi1coval 22668 pi1coghm 22669 vitalilem3 23185 ismntoplly 29397 linedegen 31420 fvline 31421 |
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