Theorem un2122 42299

Description: A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
un2122.1	⊢ (((𝜑 ∧ 𝜓) ∧ 𝜓 ∧ 𝜓) → 𝜒)

Assertion

Ref	Expression
un2122	⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Proof of Theorem un2122

Step	Hyp	Ref	Expression
1		3anass 1093	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜓 ∧ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜓 ∧ 𝜓)))
2		anandir 673	. . . 4 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜓 ∧ 𝜓)))
3		ancom 460	. . . . 5 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜓) ↔ (𝜓 ∧ (𝜑 ∧ 𝜓)))
4		anabs7 660	. . . . 5 ⊢ ((𝜓 ∧ (𝜑 ∧ 𝜓)) ↔ (𝜑 ∧ 𝜓))
5	3, 4	bitri 274	. . . 4 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜓) ↔ (𝜑 ∧ 𝜓))
6	2, 5	bitr3i 276	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜓 ∧ 𝜓)) ↔ (𝜑 ∧ 𝜓))
7	1, 6	bitri 274	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜓 ∧ 𝜓) ↔ (𝜑 ∧ 𝜓))
8		un2122.1	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜓 ∧ 𝜓) → 𝜒)
9	7, 8	sylbir 234	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085

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

This theorem is referenced by:  suctrALT3  42433