Description: An alternate definition of the conditional operator df-if as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013)
Ref | Expression | ||
---|---|---|---|
Assertion | dfif6 | ⊢ if ( 𝜑 , 𝐴 , 𝐵 ) = ( { 𝑥 ∈ 𝐴 ∣ 𝜑 } ∪ { 𝑥 ∈ 𝐵 ∣ ¬ 𝜑 } ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unab | ⊢ ( { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } ∪ { 𝑥 ∣ ( 𝑥 ∈ 𝐵 ∧ ¬ 𝜑 ) } ) = { 𝑥 ∣ ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∨ ( 𝑥 ∈ 𝐵 ∧ ¬ 𝜑 ) ) } | |
2 | df-rab | ⊢ { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } | |
3 | df-rab | ⊢ { 𝑥 ∈ 𝐵 ∣ ¬ 𝜑 } = { 𝑥 ∣ ( 𝑥 ∈ 𝐵 ∧ ¬ 𝜑 ) } | |
4 | 2 3 | uneq12i | ⊢ ( { 𝑥 ∈ 𝐴 ∣ 𝜑 } ∪ { 𝑥 ∈ 𝐵 ∣ ¬ 𝜑 } ) = ( { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } ∪ { 𝑥 ∣ ( 𝑥 ∈ 𝐵 ∧ ¬ 𝜑 ) } ) |
5 | df-if | ⊢ if ( 𝜑 , 𝐴 , 𝐵 ) = { 𝑥 ∣ ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∨ ( 𝑥 ∈ 𝐵 ∧ ¬ 𝜑 ) ) } | |
6 | 1 4 5 | 3eqtr4ri | ⊢ if ( 𝜑 , 𝐴 , 𝐵 ) = ( { 𝑥 ∈ 𝐴 ∣ 𝜑 } ∪ { 𝑥 ∈ 𝐵 ∣ ¬ 𝜑 } ) |