Description: Equivalence/equality deduction for conditional operators. (Contributed by JJ, 25-Sep-2018)
Ref | Expression | ||
---|---|---|---|
Hypotheses | ifbieq1d.1 | ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) | |
ifbieq1d.2 | ⊢ ( 𝜑 → 𝐴 = 𝐵 ) | ||
Assertion | ifbieq1d | ⊢ ( 𝜑 → if ( 𝜓 , 𝐴 , 𝐶 ) = if ( 𝜒 , 𝐵 , 𝐶 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifbieq1d.1 | ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) | |
2 | ifbieq1d.2 | ⊢ ( 𝜑 → 𝐴 = 𝐵 ) | |
3 | 1 | ifbid | ⊢ ( 𝜑 → if ( 𝜓 , 𝐴 , 𝐶 ) = if ( 𝜒 , 𝐴 , 𝐶 ) ) |
4 | 2 | ifeq1d | ⊢ ( 𝜑 → if ( 𝜒 , 𝐴 , 𝐶 ) = if ( 𝜒 , 𝐵 , 𝐶 ) ) |
5 | 3 4 | eqtrd | ⊢ ( 𝜑 → if ( 𝜓 , 𝐴 , 𝐶 ) = if ( 𝜒 , 𝐵 , 𝐶 ) ) |