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Mirrors > Home > MPE Home > Th. List > qseq2 | Structured version Visualization version GIF version |
Description: Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.) |
Ref | Expression |
---|---|
qseq2 | ⊢ (𝐴 = 𝐵 → (𝐶 / 𝐴) = (𝐶 / 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eceq2 8689 | . . . . 5 ⊢ (𝐴 = 𝐵 → [𝑥]𝐴 = [𝑥]𝐵) | |
2 | 1 | eqeq2d 2748 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝑦 = [𝑥]𝐴 ↔ 𝑦 = [𝑥]𝐵)) |
3 | 2 | rexbidv 3176 | . . 3 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐶 𝑦 = [𝑥]𝐴 ↔ ∃𝑥 ∈ 𝐶 𝑦 = [𝑥]𝐵)) |
4 | 3 | abbidv 2806 | . 2 ⊢ (𝐴 = 𝐵 → {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑦 = [𝑥]𝐴} = {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑦 = [𝑥]𝐵}) |
5 | df-qs 8655 | . 2 ⊢ (𝐶 / 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑦 = [𝑥]𝐴} | |
6 | df-qs 8655 | . 2 ⊢ (𝐶 / 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑦 = [𝑥]𝐵} | |
7 | 4, 5, 6 | 3eqtr4g 2802 | 1 ⊢ (𝐴 = 𝐵 → (𝐶 / 𝐴) = (𝐶 / 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 {cab 2714 ∃wrex 3074 [cec 8647 / cqs 8648 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-cnv 5642 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-ec 8651 df-qs 8655 |
This theorem is referenced by: qseq2i 8705 qseq2d 8706 qseq12 8707 pi1bas3 24409 |
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