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Mirrors > Home > ILE Home > Th. List > ifbi | GIF version |
Description: Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.) |
Ref | Expression |
---|---|
ifbi | ⊢ ((𝜑 ↔ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | anbi2 467 | . . . 4 ⊢ ((𝜑 ↔ 𝜓) → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ 𝜓))) | |
2 | notbi 667 | . . . . 5 ⊢ ((𝜑 ↔ 𝜓) → (¬ 𝜑 ↔ ¬ 𝜓)) | |
3 | 2 | anbi2d 464 | . . . 4 ⊢ ((𝜑 ↔ 𝜓) → ((𝑥 ∈ 𝐵 ∧ ¬ 𝜑) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝜓))) |
4 | 1, 3 | orbi12d 794 | . . 3 ⊢ ((𝜑 ↔ 𝜓) → (((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜓) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜓)))) |
5 | 4 | abbidv 2307 | . 2 ⊢ ((𝜑 ↔ 𝜓) → {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))} = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜓) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜓))}) |
6 | df-if 3550 | . 2 ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))} | |
7 | df-if 3550 | . 2 ⊢ if(𝜓, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜓) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜓))} | |
8 | 5, 6, 7 | 3eqtr4g 2247 | 1 ⊢ ((𝜑 ↔ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 709 = wceq 1364 ∈ wcel 2160 {cab 2175 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-in1 615 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-11 1517 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-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-if 3550 |
This theorem is referenced by: ifbid 3570 ifbieq2i 3572 fodjuomni 7177 fodjumkv 7188 nninfwlpoimlemg 7203 1tonninf 10471 lgsdi 14899 nninfsellemqall 15226 nninfomni 15230 |
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