Mathbox for BJ |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-dfif | Structured version Visualization version GIF version |
Description: Alternate definition of the conditional operator for classes, which used to be the main definition. (Contributed by BJ, 26-Dec-2023.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-dfif | ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝜑 ∧ 𝑥 ∈ 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 ∈ 𝐵))} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-df-ifc 34498 | . 2 ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵)} | |
2 | df-ifp 1064 | . . 3 ⊢ (if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵) ↔ ((𝜑 ∧ 𝑥 ∈ 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 ∈ 𝐵))) | |
3 | 2 | abbii 2808 | . 2 ⊢ {𝑥 ∣ if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵)} = {𝑥 ∣ ((𝜑 ∧ 𝑥 ∈ 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 ∈ 𝐵))} |
4 | 1, 3 | eqtri 2765 | 1 ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝜑 ∧ 𝑥 ∈ 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 ∈ 𝐵))} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 399 ∨ wo 847 if-wif 1063 = wceq 1543 ∈ wcel 2110 {cab 2714 ifcif 4439 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-ifp 1064 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-if 4440 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |