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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 5923 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 “ {𝐶}) = (𝐵 “ {𝐶})) | |
2 | df-ec 8290 | . 2 ⊢ [𝐶]𝐴 = (𝐴 “ {𝐶}) | |
3 | df-ec 8290 | . 2 ⊢ [𝐶]𝐵 = (𝐵 “ {𝐶}) | |
4 | 1, 2, 3 | 3eqtr4g 2881 | 1 ⊢ (𝐴 = 𝐵 → [𝐶]𝐴 = [𝐶]𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 {csn 4566 “ cima 5557 [cec 8286 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-br 5066 df-opab 5128 df-cnv 5562 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-ec 8290 |
This theorem is referenced by: eceq2i 8329 eceq2d 8330 qseq2 8343 qusval 16814 efgrelexlemb 18875 efgcpbllemb 18880 znzrh2 20691 |
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