Description: Equality deduction for well-orders. (Contributed by Stefan O'Rear, 19-Jan-2015)
Ref | Expression | ||
---|---|---|---|
Hypotheses | weeq12d.l | ⊢ ( 𝜑 → 𝑅 = 𝑆 ) | |
weeq12d.r | ⊢ ( 𝜑 → 𝐴 = 𝐵 ) | ||
Assertion | weeq12d | ⊢ ( 𝜑 → ( 𝑅 We 𝐴 ↔ 𝑆 We 𝐵 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | weeq12d.l | ⊢ ( 𝜑 → 𝑅 = 𝑆 ) | |
2 | weeq12d.r | ⊢ ( 𝜑 → 𝐴 = 𝐵 ) | |
3 | weeq1 | ⊢ ( 𝑅 = 𝑆 → ( 𝑅 We 𝐴 ↔ 𝑆 We 𝐴 ) ) | |
4 | 1 3 | syl | ⊢ ( 𝜑 → ( 𝑅 We 𝐴 ↔ 𝑆 We 𝐴 ) ) |
5 | weeq2 | ⊢ ( 𝐴 = 𝐵 → ( 𝑆 We 𝐴 ↔ 𝑆 We 𝐵 ) ) | |
6 | 2 5 | syl | ⊢ ( 𝜑 → ( 𝑆 We 𝐴 ↔ 𝑆 We 𝐵 ) ) |
7 | 4 6 | bitrd | ⊢ ( 𝜑 → ( 𝑅 We 𝐴 ↔ 𝑆 We 𝐵 ) ) |