Theorem uunT12p3 42311

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
uunT12p3.1	⊢ ((𝜓 ∧ ⊤ ∧ 𝜑) → 𝜒)

Assertion

Ref	Expression
uunT12p3	⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Proof of Theorem uunT12p3

Step	Hyp	Ref	Expression
1		3ancoma 1096	. . . . 5 ⊢ ((𝜓 ∧ ⊤ ∧ 𝜑) ↔ (⊤ ∧ 𝜓 ∧ 𝜑))
2		3anass 1093	. . . . 5 ⊢ ((⊤ ∧ 𝜓 ∧ 𝜑) ↔ (⊤ ∧ (𝜓 ∧ 𝜑)))
3	1, 2	bitri 274	. . . 4 ⊢ ((𝜓 ∧ ⊤ ∧ 𝜑) ↔ (⊤ ∧ (𝜓 ∧ 𝜑)))
4		truan 1550	. . . 4 ⊢ ((⊤ ∧ (𝜓 ∧ 𝜑)) ↔ (𝜓 ∧ 𝜑))
5	3, 4	bitri 274	. . 3 ⊢ ((𝜓 ∧ ⊤ ∧ 𝜑) ↔ (𝜓 ∧ 𝜑))
6		ancom 460	. . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑))
7	5, 6	bitr4i 277	. 2 ⊢ ((𝜓 ∧ ⊤ ∧ 𝜑) ↔ (𝜑 ∧ 𝜓))
8		uunT12p3.1	. 2 ⊢ ((𝜓 ∧ ⊤ ∧ 𝜑) → 𝜒)
9	7, 8	sylbir 234	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085  ⊤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 206  df-an 396  df-3an 1087  df-tru 1542

This theorem is referenced by: (None)