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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 3880 | . 2 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐶 / 𝑥⦌𝐵) |
3 | csbeq12dv.2 | . . 3 ⊢ (𝜑 → 𝐵 = 𝐷) | |
4 | 3 | csbeq2dv 3883 | . 2 ⊢ (𝜑 → ⦋𝐶 / 𝑥⦌𝐵 = ⦋𝐶 / 𝑥⦌𝐷) |
5 | 2, 4 | eqtrd 2855 | 1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐶 / 𝑥⦌𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ⦋csb 3876 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-sbc 3769 df-csb 3877 |
This theorem is referenced by: selvffval 20322 selvfval 20323 selvval 20324 |
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