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Mirrors > Home > MPE Home > Th. List > ifbieq12d2 | Structured version Visualization version GIF version |
Description: Equivalence deduction for conditional operators. (Contributed by Thierry Arnoux, 14-Feb-2017.) (Proof shortened by Wolf Lammen, 24-Jun-2021.) |
Ref | Expression |
---|---|
ifbieq12d2.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
ifbieq12d2.2 | ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶) |
ifbieq12d2.3 | ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐷) |
Ref | Expression |
---|---|
ifbieq12d2 | ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifbieq12d2.2 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶) | |
2 | ifbieq12d2.3 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐷) | |
3 | 1, 2 | ifeq12da 4118 | . 2 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷)) |
4 | ifbieq12d2.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
5 | 4 | ifbid 4108 | . 2 ⊢ (𝜑 → if(𝜓, 𝐶, 𝐷) = if(𝜒, 𝐶, 𝐷)) |
6 | 3, 5 | eqtrd 2656 | 1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ifcif 4086 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-un 3579 df-if 4087 |
This theorem is referenced by: ofccat 13708 itgeq12dv 30388 sgnneg 30602 |
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