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Mirrors > Home > MPE Home > Th. List > eceq2 | Structured version Visualization version GIF version |
Description: Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.) |
Ref | Expression |
---|---|
eceq2 | ⊢ (𝐴 = 𝐵 → [𝐶]𝐴 = [𝐶]𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imaeq1 6054 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 “ {𝐶}) = (𝐵 “ {𝐶})) | |
2 | df-ec 8711 | . 2 ⊢ [𝐶]𝐴 = (𝐴 “ {𝐶}) | |
3 | df-ec 8711 | . 2 ⊢ [𝐶]𝐵 = (𝐵 “ {𝐶}) | |
4 | 1, 2, 3 | 3eqtr4g 2796 | 1 ⊢ (𝐴 = 𝐵 → [𝐶]𝐴 = [𝐶]𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 {csn 4628 “ cima 5679 [cec 8707 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-cnv 5684 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-ec 8711 |
This theorem is referenced by: eceq2i 8750 eceq2d 8751 qseq2 8764 qusval 17495 efgrelexlemb 19666 efgcpbllemb 19671 znzrh2 21411 |
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