Description: Deduction joining two equivalences to form equivalence of implications. (Contributed by NM, 16-May-1993)
Ref | Expression | ||
---|---|---|---|
Hypotheses | imbi12d.1 | ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) | |
imbi12d.2 | ⊢ ( 𝜑 → ( 𝜃 ↔ 𝜏 ) ) | ||
Assertion | imbi12d | ⊢ ( 𝜑 → ( ( 𝜓 → 𝜃 ) ↔ ( 𝜒 → 𝜏 ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imbi12d.1 | ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) | |
2 | imbi12d.2 | ⊢ ( 𝜑 → ( 𝜃 ↔ 𝜏 ) ) | |
3 | 1 | imbi1d | ⊢ ( 𝜑 → ( ( 𝜓 → 𝜃 ) ↔ ( 𝜒 → 𝜃 ) ) ) |
4 | 2 | imbi2d | ⊢ ( 𝜑 → ( ( 𝜒 → 𝜃 ) ↔ ( 𝜒 → 𝜏 ) ) ) |
5 | 3 4 | bitrd | ⊢ ( 𝜑 → ( ( 𝜓 → 𝜃 ) ↔ ( 𝜒 → 𝜏 ) ) ) |