Theorem uun2221

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.)

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

Proof

Step	Hyp	Ref	Expression
1		uun2221.1	⊢ ( ( 𝜑 ∧ 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) → 𝜒 )
2		3anass	⊢ ( ( 𝜑 ∧ 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) ↔ ( 𝜑 ∧ ( 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) ) )
3		anabs5	⊢ ( ( 𝜑 ∧ ( 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) ) ↔ ( 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) )
4	2 3	bitri	⊢ ( ( 𝜑 ∧ 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) ↔ ( 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) )
5		ancom	⊢ ( ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜓 ∧ 𝜑 ) )
6	5	anbi2i	⊢ ( ( 𝜑 ∧ ( 𝜑 ∧ 𝜓 ) ) ↔ ( 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) )
7	4 6	bitr4i	⊢ ( ( 𝜑 ∧ 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) ↔ ( 𝜑 ∧ ( 𝜑 ∧ 𝜓 ) ) )
8		anabs5	⊢ ( ( 𝜑 ∧ ( 𝜑 ∧ 𝜓 ) ) ↔ ( 𝜑 ∧ 𝜓 ) )
9	8 5	bitri	⊢ ( ( 𝜑 ∧ ( 𝜑 ∧ 𝜓 ) ) ↔ ( 𝜓 ∧ 𝜑 ) )
10	7 9	bitri	⊢ ( ( 𝜑 ∧ 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) ↔ ( 𝜓 ∧ 𝜑 ) )
11	10	imbi1i	⊢ ( ( ( 𝜑 ∧ 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) → 𝜒 ) ↔ ( ( 𝜓 ∧ 𝜑 ) → 𝜒 ) )
12	1 11	mpbi	⊢ ( ( 𝜓 ∧ 𝜑 ) → 𝜒 )