Description: Equivalence deduction for conditional operators. (Contributed by Jeff Madsen, 2-Sep-2009)
Ref | Expression | ||
---|---|---|---|
Hypotheses | ifbieq12d.1 | ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) | |
ifbieq12d.2 | ⊢ ( 𝜑 → 𝐴 = 𝐶 ) | ||
ifbieq12d.3 | ⊢ ( 𝜑 → 𝐵 = 𝐷 ) | ||
Assertion | ifbieq12d | ⊢ ( 𝜑 → if ( 𝜓 , 𝐴 , 𝐵 ) = if ( 𝜒 , 𝐶 , 𝐷 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifbieq12d.1 | ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) | |
2 | ifbieq12d.2 | ⊢ ( 𝜑 → 𝐴 = 𝐶 ) | |
3 | ifbieq12d.3 | ⊢ ( 𝜑 → 𝐵 = 𝐷 ) | |
4 | 1 | ifbid | ⊢ ( 𝜑 → if ( 𝜓 , 𝐴 , 𝐵 ) = if ( 𝜒 , 𝐴 , 𝐵 ) ) |
5 | 2 3 | ifeq12d | ⊢ ( 𝜑 → if ( 𝜒 , 𝐴 , 𝐵 ) = if ( 𝜒 , 𝐶 , 𝐷 ) ) |
6 | 4 5 | eqtrd | ⊢ ( 𝜑 → if ( 𝜓 , 𝐴 , 𝐵 ) = if ( 𝜒 , 𝐶 , 𝐷 ) ) |