Theorem uunT12p5 44829

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.)

Hypothesis

Ref	Expression
uunT12p5.1	⊢ ((𝜓 ∧ 𝜑 ∧ ⊤) → 𝜒)

Assertion

Ref	Expression
uunT12p5	⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Proof of Theorem uunT12p5

Step	Hyp	Ref	Expression
1		3anrev 1100	. . . 4 ⊢ ((𝜓 ∧ 𝜑 ∧ ⊤) ↔ (⊤ ∧ 𝜑 ∧ 𝜓))
2		3anass 1094	. . . 4 ⊢ ((⊤ ∧ 𝜑 ∧ 𝜓) ↔ (⊤ ∧ (𝜑 ∧ 𝜓)))
3	1, 2	bitri 275	. . 3 ⊢ ((𝜓 ∧ 𝜑 ∧ ⊤) ↔ (⊤ ∧ (𝜑 ∧ 𝜓)))
4		truan 1550	. . 3 ⊢ ((⊤ ∧ (𝜑 ∧ 𝜓)) ↔ (𝜑 ∧ 𝜓))
5	3, 4	bitri 275	. 2 ⊢ ((𝜓 ∧ 𝜑 ∧ ⊤) ↔ (𝜑 ∧ 𝜓))
6		uunT12p5.1	. 2 ⊢ ((𝜓 ∧ 𝜑 ∧ ⊤) → 𝜒)
7	5, 6	sylbir 235	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086  ⊤wtru 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1542

This theorem is referenced by: (None)