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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 456 | . . . 4 ⊢ ((𝜑 ↔ 𝜓) → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ 𝜓))) | |
2 | id 19 | . . . . . 6 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 ↔ 𝜓)) | |
3 | 2 | notbid 630 | . . . . 5 ⊢ ((𝜑 ↔ 𝜓) → (¬ 𝜑 ↔ ¬ 𝜓)) |
4 | 3 | anbi2d 453 | . . . 4 ⊢ ((𝜑 ↔ 𝜓) → ((𝑥 ∈ 𝐵 ∧ ¬ 𝜑) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝜓))) |
5 | 1, 4 | orbi12d 745 | . . 3 ⊢ ((𝜑 ↔ 𝜓) → (((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜓) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜓)))) |
6 | 5 | abbidv 2212 | . 2 ⊢ ((𝜑 ↔ 𝜓) → {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))} = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜓) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜓))}) |
7 | df-if 3414 | . 2 ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))} | |
8 | df-if 3414 | . 2 ⊢ if(𝜓, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜓) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜓))} | |
9 | 6, 7, 8 | 3eqtr4g 2152 | 1 ⊢ ((𝜑 ↔ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 667 = wceq 1296 ∈ wcel 1445 {cab 2081 ifcif 3413 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-11 1449 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-if 3414 |
This theorem is referenced by: ifbid 3432 ifbieq2i 3434 fodjuomni 6892 fodjumkv 6903 1tonninf 9995 nninfsellemqall 12616 nninfomni 12620 |
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