Theorem uunT12p2 42035

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
uunT12p2.1	⊢ ((𝜑 ∧ ⊤ ∧ 𝜓) → 𝜒)

Assertion

Ref	Expression
uunT12p2	⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Proof of Theorem uunT12p2

Step	Hyp	Ref	Expression
1		3anrot 1102	. . . . 5 ⊢ ((𝜑 ∧ ⊤ ∧ 𝜓) ↔ (⊤ ∧ 𝜓 ∧ 𝜑))
2		3anass 1097	. . . . 5 ⊢ ((⊤ ∧ 𝜓 ∧ 𝜑) ↔ (⊤ ∧ (𝜓 ∧ 𝜑)))
3	1, 2	bitri 278	. . . 4 ⊢ ((𝜑 ∧ ⊤ ∧ 𝜓) ↔ (⊤ ∧ (𝜓 ∧ 𝜑)))
4		truan 1554	. . . 4 ⊢ ((⊤ ∧ (𝜓 ∧ 𝜑)) ↔ (𝜓 ∧ 𝜑))
5	3, 4	bitri 278	. . 3 ⊢ ((𝜑 ∧ ⊤ ∧ 𝜓) ↔ (𝜓 ∧ 𝜑))
6		ancom 464	. . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑))
7	5, 6	bitr4i 281	. 2 ⊢ ((𝜑 ∧ ⊤ ∧ 𝜓) ↔ (𝜑 ∧ 𝜓))
8		uunT12p2.1	. 2 ⊢ ((𝜑 ∧ ⊤ ∧ 𝜓) → 𝜒)
9	7, 8	sylbir 238	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1089  ⊤wtru 1544

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 210  df-an 400  df-3an 1091  df-tru 1546

This theorem is referenced by: (None)