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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 5380 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 “ {𝐶}) = (𝐵 “ {𝐶})) | |
2 | df-ec 7631 | . 2 ⊢ [𝐶]𝐴 = (𝐴 “ {𝐶}) | |
3 | df-ec 7631 | . 2 ⊢ [𝐶]𝐵 = (𝐵 “ {𝐶}) | |
4 | 1, 2, 3 | 3eqtr4g 2669 | 1 ⊢ (𝐴 = 𝐵 → [𝐶]𝐴 = [𝐶]𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 {csn 4125 “ cima 5041 [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-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ec 7631 |
This theorem is referenced by: qseq2 7684 qusval 16025 efgrelexlemb 17986 efgcpbllemb 17991 vrgpfval 18002 znzrh2 19713 |
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