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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 3912 | . 2 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐶 / 𝑥⦌𝐵) |
3 | csbeq12dv.2 | . . 3 ⊢ (𝜑 → 𝐵 = 𝐷) | |
4 | 3 | csbeq2dv 3915 | . 2 ⊢ (𝜑 → ⦋𝐶 / 𝑥⦌𝐵 = ⦋𝐶 / 𝑥⦌𝐷) |
5 | 2, 4 | eqtrd 2775 | 1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐶 / 𝑥⦌𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ⦋csb 3908 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-sbc 3792 df-csb 3909 |
This theorem is referenced by: bpolylem 16081 selvffval 22155 selvfval 22156 selvval 22157 cbvitgv 25827 mulsval 28150 precsexlemcbv 28245 precsexlem3 28248 ttgval 28898 itgeq12sdv 36202 cbvitgvw2 36231 cbvitgdavw 36264 cbvitgdavw2 36280 poimirlem16 37623 poimirlem17 37624 poimirlem19 37626 poimirlem20 37627 isprimroot 42075 fmpocos 42254 grtri 47845 |
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