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Mirrors > Home > ILE Home > Th. List > ifeq2dadc | GIF version |
Description: Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
ifeq2da.1 | ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐴 = 𝐵) |
ifeq2dadc.dc | ⊢ (𝜑 → DECID 𝜓) |
Ref | Expression |
---|---|
ifeq2dadc | ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 110 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝜓) | |
2 | 1 | iftrued 3556 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐶, 𝐴) = 𝐶) |
3 | 1 | iftrued 3556 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐶, 𝐵) = 𝐶) |
4 | 2, 3 | eqtr4d 2225 | . 2 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)) |
5 | ifeq2da.1 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐴 = 𝐵) | |
6 | 5 | ifeq2d 3567 | . 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)) |
7 | ifeq2dadc.dc | . . 3 ⊢ (𝜑 → DECID 𝜓) | |
8 | exmiddc 837 | . . 3 ⊢ (DECID 𝜓 → (𝜓 ∨ ¬ 𝜓)) | |
9 | 7, 8 | syl 14 | . 2 ⊢ (𝜑 → (𝜓 ∨ ¬ 𝜓)) |
10 | 4, 6, 9 | mpjaodan 799 | 1 ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ∨ wo 709 DECID wdc 835 = wceq 1364 ifcif 3549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-rab 2477 df-v 2754 df-un 3148 df-if 3550 |
This theorem is referenced by: subgmulg 13099 |
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