Metamath Proof Explorer
Description: Join two logical equivalences with anti-conjunction. (Contributed by Scott Fenton, 2-Jan-2018)
|
|
Ref |
Expression |
|
Hypotheses |
nanbid.1 |
⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) |
|
|
nanbi12d.2 |
⊢ ( 𝜑 → ( 𝜃 ↔ 𝜏 ) ) |
|
Assertion |
nanbi12d |
⊢ ( 𝜑 → ( ( 𝜓 ⊼ 𝜃 ) ↔ ( 𝜒 ⊼ 𝜏 ) ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
nanbid.1 |
⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) |
2 |
|
nanbi12d.2 |
⊢ ( 𝜑 → ( 𝜃 ↔ 𝜏 ) ) |
3 |
|
nanbi12 |
⊢ ( ( ( 𝜓 ↔ 𝜒 ) ∧ ( 𝜃 ↔ 𝜏 ) ) → ( ( 𝜓 ⊼ 𝜃 ) ↔ ( 𝜒 ⊼ 𝜏 ) ) ) |
4 |
1 2 3
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝜓 ⊼ 𝜃 ) ↔ ( 𝜒 ⊼ 𝜏 ) ) ) |