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Metamath Proof Explorer
Description: Distribution of implication over biconditional (deduction form).
Variant of pm5.74d . (Contributed by NM, 4-May-2007)
|
|
Ref |
Expression |
|
Hypothesis |
pm5.74da.1 |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜒 ↔ 𝜃 ) ) |
|
Assertion |
pm5.74da |
⊢ ( 𝜑 → ( ( 𝜓 → 𝜒 ) ↔ ( 𝜓 → 𝜃 ) ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
pm5.74da.1 |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜒 ↔ 𝜃 ) ) |
| 2 |
1
|
ex |
⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 ↔ 𝜃 ) ) ) |
| 3 |
2
|
pm5.74d |
⊢ ( 𝜑 → ( ( 𝜓 → 𝜒 ) ↔ ( 𝜓 → 𝜃 ) ) ) |