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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 5899 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 “ {𝐶}) = (𝐵 “ {𝐶})) | |
2 | df-ec 8325 | . 2 ⊢ [𝐶]𝐴 = (𝐴 “ {𝐶}) | |
3 | df-ec 8325 | . 2 ⊢ [𝐶]𝐵 = (𝐵 “ {𝐶}) | |
4 | 1, 2, 3 | 3eqtr4g 2799 | 1 ⊢ (𝐴 = 𝐵 → [𝐶]𝐴 = [𝐶]𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 {csn 4517 “ cima 5529 [cec 8321 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2711 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-ex 1787 df-sb 2075 df-clab 2718 df-cleq 2731 df-clel 2812 df-rab 3063 df-v 3401 df-un 3849 df-in 3851 df-ss 3861 df-sn 4518 df-pr 4520 df-op 4524 df-br 5032 df-opab 5094 df-cnv 5534 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-ec 8325 |
This theorem is referenced by: eceq2i 8364 eceq2d 8365 qseq2 8378 qusval 16921 efgrelexlemb 18997 efgcpbllemb 19002 znzrh2 20367 |
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