Description: A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017) (Proof modification is discouraged.) (New usage is discouraged.)
Ref | Expression | ||
---|---|---|---|
Hypothesis | uunT12p3.1 | ⊢ ( ( 𝜓 ∧ ⊤ ∧ 𝜑 ) → 𝜒 ) | |
Assertion | uunT12p3 | ⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uunT12p3.1 | ⊢ ( ( 𝜓 ∧ ⊤ ∧ 𝜑 ) → 𝜒 ) | |
2 | 3ancoma | ⊢ ( ( 𝜓 ∧ ⊤ ∧ 𝜑 ) ↔ ( ⊤ ∧ 𝜓 ∧ 𝜑 ) ) | |
3 | 3anass | ⊢ ( ( ⊤ ∧ 𝜓 ∧ 𝜑 ) ↔ ( ⊤ ∧ ( 𝜓 ∧ 𝜑 ) ) ) | |
4 | 2 3 | bitri | ⊢ ( ( 𝜓 ∧ ⊤ ∧ 𝜑 ) ↔ ( ⊤ ∧ ( 𝜓 ∧ 𝜑 ) ) ) |
5 | truan | ⊢ ( ( ⊤ ∧ ( 𝜓 ∧ 𝜑 ) ) ↔ ( 𝜓 ∧ 𝜑 ) ) | |
6 | 4 5 | bitri | ⊢ ( ( 𝜓 ∧ ⊤ ∧ 𝜑 ) ↔ ( 𝜓 ∧ 𝜑 ) ) |
7 | ancom | ⊢ ( ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜓 ∧ 𝜑 ) ) | |
8 | 6 7 | bitr4i | ⊢ ( ( 𝜓 ∧ ⊤ ∧ 𝜑 ) ↔ ( 𝜑 ∧ 𝜓 ) ) |
9 | 8 1 | sylbir | ⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 ) |