Metamath Proof Explorer


Theorem uunT12p3

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 ( ( 𝜑𝜓 ) → 𝜒 )

Proof

Step Hyp Ref Expression
1 uunT12p3.1 ( ( 𝜓 ∧ ⊤ ∧ 𝜑 ) → 𝜒 )
2 3ancoma ( ( 𝜓 ∧ ⊤ ∧ 𝜑 ) ↔ ( ⊤ ∧ 𝜓𝜑 ) )
3 3anass ( ( ⊤ ∧ 𝜓𝜑 ) ↔ ( ⊤ ∧ ( 𝜓𝜑 ) ) )
4 2 3 bitri ( ( 𝜓 ∧ ⊤ ∧ 𝜑 ) ↔ ( ⊤ ∧ ( 𝜓𝜑 ) ) )
5 truan ( ( ⊤ ∧ ( 𝜓𝜑 ) ) ↔ ( 𝜓𝜑 ) )
6 4 5 bitri ( ( 𝜓 ∧ ⊤ ∧ 𝜑 ) ↔ ( 𝜓𝜑 ) )
7 ancom ( ( 𝜑𝜓 ) ↔ ( 𝜓𝜑 ) )
8 6 7 bitr4i ( ( 𝜓 ∧ ⊤ ∧ 𝜑 ) ↔ ( 𝜑𝜓 ) )
9 8 1 sylbir ( ( 𝜑𝜓 ) → 𝜒 )