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Mirrors > Home > MPE Home > Th. List > ifeq12da | Structured version Visualization version GIF version |
Description: Equivalence deduction for conditional operators. (Contributed by Wolf Lammen, 24-Jun-2021.) |
Ref | Expression |
---|---|
ifeq12da.1 | ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶) |
ifeq12da.2 | ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐷) |
Ref | Expression |
---|---|
ifeq12da | ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifeq12da.1 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶) | |
2 | 1 | ifeq1da 4493 | . . 3 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐵)) |
3 | iftrue 4469 | . . . 4 ⊢ (𝜓 → if(𝜓, 𝐶, 𝐵) = 𝐶) | |
4 | iftrue 4469 | . . . 4 ⊢ (𝜓 → if(𝜓, 𝐶, 𝐷) = 𝐶) | |
5 | 3, 4 | eqtr4d 2856 | . . 3 ⊢ (𝜓 → if(𝜓, 𝐶, 𝐵) = if(𝜓, 𝐶, 𝐷)) |
6 | 2, 5 | sylan9eq 2873 | . 2 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷)) |
7 | ifeq12da.2 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐷) | |
8 | 7 | ifeq2da 4494 | . . 3 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐷)) |
9 | iffalse 4472 | . . . 4 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐷) = 𝐷) | |
10 | iffalse 4472 | . . . 4 ⊢ (¬ 𝜓 → if(𝜓, 𝐶, 𝐷) = 𝐷) | |
11 | 9, 10 | eqtr4d 2856 | . . 3 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐷) = if(𝜓, 𝐶, 𝐷)) |
12 | 8, 11 | sylan9eq 2873 | . 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷)) |
13 | 6, 12 | pm2.61dan 809 | 1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1528 ifcif 4463 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-un 3938 df-if 4464 |
This theorem is referenced by: ifbieq12d2 4496 copco 23549 pcohtpylem 23550 rpvmasum2 26015 |
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