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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 3791 | . . 3 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝐵 = 𝐶 ↔ [𝐴 / 𝑥]𝐵 = 𝐶)) | |
| 2 | dfsbcq2 3791 | . . . . 5 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝑦 ∈ 𝐵 ↔ [𝐴 / 𝑥]𝑦 ∈ 𝐵)) | |
| 3 | 2 | abbidv 2808 | . . . 4 ⊢ (𝑧 = 𝐴 → {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}) |
| 4 | dfsbcq2 3791 | . . . . 5 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝑦 ∈ 𝐶 ↔ [𝐴 / 𝑥]𝑦 ∈ 𝐶)) | |
| 5 | 4 | abbidv 2808 | . . . 4 ⊢ (𝑧 = 𝐴 → {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶}) |
| 6 | 3, 5 | eqeq12d 2753 | . . 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 2156 | . . . . . 6 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝑦 ∈ 𝐵 | |
| 8 | 7 | nfab 2911 | . . . . 5 ⊢ Ⅎ𝑥{𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵} |
| 9 | nfs1v 2156 | . . . . . 6 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝑦 ∈ 𝐶 | |
| 10 | 9 | nfab 2911 | . . . . 5 ⊢ Ⅎ𝑥{𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶} |
| 11 | 8, 10 | nfeq 2919 | . . . 4 ⊢ Ⅎ𝑥{𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶} |
| 12 | sbab 2889 | . . . . 5 ⊢ (𝑥 = 𝑧 → 𝐵 = {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵}) | |
| 13 | sbab 2889 | . . . . 5 ⊢ (𝑥 = 𝑧 → 𝐶 = {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶}) | |
| 14 | 12, 13 | eqeq12d 2753 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝐵 = 𝐶 ↔ {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶})) |
| 15 | 11, 14 | sbiev 2314 | . . 3 ⊢ ([𝑧 / 𝑥]𝐵 = 𝐶 ↔ {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ [𝑧 / 𝑥]𝑦 ∈ 𝐶}) |
| 16 | 1, 6, 15 | vtoclbg 3557 | . 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶})) |
| 17 | df-csb 3900 | . . 3 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} | |
| 18 | df-csb 3900 | . . 3 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶} | |
| 19 | 17, 18 | eqeq12i 2755 | . 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 206 = wceq 1540 [wsb 2064 ∈ wcel 2108 {cab 2714 [wsbc 3788 ⦋csb 3899 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-sbc 3789 df-csb 3900 |
| This theorem is referenced by: sbceqi 4413 sbcne12 4415 sbceq1g 4417 sbceq2g 4419 csbie2df 4443 sbcfng 6733 csbfrecsg 8309 swrdspsleq 14703 fprodmodd 16033 relowlpssretop 37365 rdgeqoa 37371 poimirlem25 37652 cdlemk42 40943 minregex 43547 onfrALTlem5 44562 onfrALTlem4 44563 csbingVD 44904 onfrALTlem5VD 44905 onfrALTlem4VD 44906 csbeq2gVD 44912 csbsngVD 44913 csbunigVD 44918 csbfv12gALTVD 44919 |
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