Theorem uun2221 41154

Description: A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 30-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
uun2221.1	⊢ ((𝜑 ∧ 𝜑 ∧ (𝜓 ∧ 𝜑)) → 𝜒)

Assertion

Ref	Expression
uun2221	⊢ ((𝜓 ∧ 𝜑) → 𝜒)

Proof of Theorem uun2221

Step	Hyp	Ref	Expression
1		uun2221.1	. 2 ⊢ ((𝜑 ∧ 𝜑 ∧ (𝜓 ∧ 𝜑)) → 𝜒)
2		3anass 1091	. . . . . 6 ⊢ ((𝜑 ∧ 𝜑 ∧ (𝜓 ∧ 𝜑)) ↔ (𝜑 ∧ (𝜑 ∧ (𝜓 ∧ 𝜑))))
3		anabs5 661	. . . . . 6 ⊢ ((𝜑 ∧ (𝜑 ∧ (𝜓 ∧ 𝜑))) ↔ (𝜑 ∧ (𝜓 ∧ 𝜑)))
4	2, 3	bitri 277	. . . . 5 ⊢ ((𝜑 ∧ 𝜑 ∧ (𝜓 ∧ 𝜑)) ↔ (𝜑 ∧ (𝜓 ∧ 𝜑)))
5		ancom 463	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑))
6	5	anbi2i 624	. . . . 5 ⊢ ((𝜑 ∧ (𝜑 ∧ 𝜓)) ↔ (𝜑 ∧ (𝜓 ∧ 𝜑)))
7	4, 6	bitr4i 280	. . . 4 ⊢ ((𝜑 ∧ 𝜑 ∧ (𝜓 ∧ 𝜑)) ↔ (𝜑 ∧ (𝜑 ∧ 𝜓)))
8		anabs5 661	. . . . 5 ⊢ ((𝜑 ∧ (𝜑 ∧ 𝜓)) ↔ (𝜑 ∧ 𝜓))
9	8, 5	bitri 277	. . . 4 ⊢ ((𝜑 ∧ (𝜑 ∧ 𝜓)) ↔ (𝜓 ∧ 𝜑))
10	7, 9	bitri 277	. . 3 ⊢ ((𝜑 ∧ 𝜑 ∧ (𝜓 ∧ 𝜑)) ↔ (𝜓 ∧ 𝜑))
11	10	imbi1i 352	. 2 ⊢ (((𝜑 ∧ 𝜑 ∧ (𝜓 ∧ 𝜑)) → 𝜒) ↔ ((𝜓 ∧ 𝜑) → 𝜒))
12	1, 11	mpbi 232	1 ⊢ ((𝜓 ∧ 𝜑) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083

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 209  df-an 399  df-3an 1085

This theorem is referenced by: (None)