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Description: Weak deduction theorem that eliminates a hypothesis ph , making it become an antecedent. We assume that a proof exists for ph when the class variable A is replaced with a specific class B . The hypothesis ch should be assigned to the inference, and the inference hypothesis eliminated with elimhyp . If the inference has other hypotheses with class variable A , these can be kept by assigning keephyp to them. For more information, see the Weak Deduction Theorem page mmdeduction.html . (Contributed by NM, 15-May-1999)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | dedth.1 | ⊢ ( 𝐴 = if ( 𝜑 , 𝐴 , 𝐵 ) → ( 𝜓 ↔ 𝜒 ) ) | |
| dedth.2 | ⊢ 𝜒 | ||
| Assertion | dedth | ⊢ ( 𝜑 → 𝜓 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dedth.1 | ⊢ ( 𝐴 = if ( 𝜑 , 𝐴 , 𝐵 ) → ( 𝜓 ↔ 𝜒 ) ) | |
| 2 | dedth.2 | ⊢ 𝜒 | |
| 3 | iftrue | ⊢ ( 𝜑 → if ( 𝜑 , 𝐴 , 𝐵 ) = 𝐴 ) | |
| 4 | 3 | eqcomd | ⊢ ( 𝜑 → 𝐴 = if ( 𝜑 , 𝐴 , 𝐵 ) ) |
| 5 | 4 1 | syl | ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) |
| 6 | 2 5 | mpbiri | ⊢ ( 𝜑 → 𝜓 ) |