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| Mirrors > Home > MPE Home > Th. List > csbeq12dv | Structured version Visualization version GIF version | ||
| Description: Formula-building inference for class substitution. (Contributed by SN, 3-Nov-2023.) |
| Ref | Expression |
|---|---|
| csbeq12dv.1 | ⊢ (𝜑 → 𝐴 = 𝐶) |
| csbeq12dv.2 | ⊢ (𝜑 → 𝐵 = 𝐷) |
| Ref | Expression |
|---|---|
| csbeq12dv | ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐶 / 𝑥⦌𝐷) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | csbeq12dv.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐶) | |
| 2 | 1 | csbeq1d 3903 | . 2 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐶 / 𝑥⦌𝐵) |
| 3 | csbeq12dv.2 | . . 3 ⊢ (𝜑 → 𝐵 = 𝐷) | |
| 4 | 3 | csbeq2dv 3906 | . 2 ⊢ (𝜑 → ⦋𝐶 / 𝑥⦌𝐵 = ⦋𝐶 / 𝑥⦌𝐷) |
| 5 | 2, 4 | eqtrd 2777 | 1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐶 / 𝑥⦌𝐷) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ⦋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-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-sbc 3789 df-csb 3900 |
| This theorem is referenced by: bpolylem 16084 selvffval 22137 selvfval 22138 selvval 22139 cbvitgv 25812 mulsval 28135 precsexlemcbv 28230 precsexlem3 28233 ttgval 28883 itgeq12sdv 36220 cbvitgvw2 36249 cbvitgdavw 36282 cbvitgdavw2 36298 poimirlem16 37643 poimirlem17 37644 poimirlem19 37646 poimirlem20 37647 isprimroot 42094 fmpocos 42275 grtri 47907 dfswapf2 48967 |
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