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Mirrors > Home > MPE Home > Th. List > sbceqg | Structured version Visualization version GIF version |
Description: Distribute proper substitution through an equality relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Ref | Expression |
---|---|
sbceqg | ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsbcq2 3743 | . . 3 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝐵 = 𝐶 ↔ [𝐴 / 𝑥]𝐵 = 𝐶)) | |
2 | dfsbcq2 3743 | . . . . 5 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝑦 ∈ 𝐵 ↔ [𝐴 / 𝑥]𝑦 ∈ 𝐵)) | |
3 | 2 | abbidv 2802 | . . . 4 ⊢ (𝑧 = 𝐴 → {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}) |
4 | dfsbcq2 3743 | . . . . 5 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝑦 ∈ 𝐶 ↔ [𝐴 / 𝑥]𝑦 ∈ 𝐶)) | |
5 | 4 | abbidv 2802 | . . . 4 ⊢ (𝑧 = 𝐴 → {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶}) |
6 | 3, 5 | eqeq12d 2749 | . . 3 ⊢ (𝑧 = 𝐴 → ({𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶} ↔ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶})) |
7 | nfs1v 2154 | . . . . . 6 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝑦 ∈ 𝐵 | |
8 | 7 | nfab 2910 | . . . . 5 ⊢ Ⅎ𝑥{𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵} |
9 | nfs1v 2154 | . . . . . 6 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝑦 ∈ 𝐶 | |
10 | 9 | nfab 2910 | . . . . 5 ⊢ Ⅎ𝑥{𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶} |
11 | 8, 10 | nfeq 2917 | . . . 4 ⊢ Ⅎ𝑥{𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶} |
12 | sbab 2883 | . . . . 5 ⊢ (𝑥 = 𝑧 → 𝐵 = {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵}) | |
13 | sbab 2883 | . . . . 5 ⊢ (𝑥 = 𝑧 → 𝐶 = {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶}) | |
14 | 12, 13 | eqeq12d 2749 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝐵 = 𝐶 ↔ {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶})) |
15 | 11, 14 | sbiev 2309 | . . 3 ⊢ ([𝑧 / 𝑥]𝐵 = 𝐶 ↔ {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶}) |
16 | 1, 6, 15 | vtoclbg 3527 | . 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶})) |
17 | df-csb 3857 | . . 3 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} | |
18 | df-csb 3857 | . . 3 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶} | |
19 | 17, 18 | eqeq12i 2751 | . 2 ⊢ (⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶 ↔ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶}) |
20 | 16, 19 | bitr4di 289 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 [wsb 2068 ∈ wcel 2107 {cab 2710 [wsbc 3740 ⦋csb 3856 |
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-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-sbc 3741 df-csb 3857 |
This theorem is referenced by: sbceqi 4371 sbcne12 4373 sbceq1g 4375 sbceq2g 4377 csbie2df 4401 sbcfng 6666 csbfrecsg 8216 swrdspsleq 14559 fprodmodd 15885 relowlpssretop 35881 rdgeqoa 35887 poimirlem25 36149 cdlemk42 39450 minregex 41894 onfrALTlem5 42912 onfrALTlem4 42913 csbingVD 43254 onfrALTlem5VD 43255 onfrALTlem4VD 43256 csbeq2gVD 43262 csbsngVD 43263 csbunigVD 43268 csbfv12gALTVD 43269 |
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