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Mathbox for Giovanni Mascellani |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > csbeq12 | Structured version Visualization version GIF version |
Description: Equality deduction for substitution in class. (Contributed by Giovanni Mascellani, 10-Apr-2018.) |
Ref | Expression |
---|---|
csbeq12 | ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 𝐶 = 𝐷) → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑥⦌𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbeq2 3816 | . 2 ⊢ (∀𝑥 𝐶 = 𝐷 → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐷) | |
2 | csbeq1 3814 | . 2 ⊢ (𝐴 = 𝐵 → ⦋𝐴 / 𝑥⦌𝐷 = ⦋𝐵 / 𝑥⦌𝐷) | |
3 | 1, 2 | sylan9eqr 2853 | 1 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 𝐶 = 𝐷) → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑥⦌𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∀wal 1520 = wceq 1522 ⦋csb 3811 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-12 2141 ax-ext 2769 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-sbc 3707 df-csb 3812 |
This theorem is referenced by: (None) |
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