Theorem uun2221p1

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.)

	Ref	Expression
Hypothesis	uun2221p1.1	⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ∧ 𝜑 ) → 𝜒 )
Assertion	uun2221p1	⊢ ( ( 𝜓 ∧ 𝜑 ) → 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		uun2221p1.1	⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ∧ 𝜑 ) → 𝜒 )
2		3anrot	⊢ ( ( 𝜑 ∧ 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) ↔ ( 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ∧ 𝜑 ) )
3	2	imbi1i	⊢ ( ( ( 𝜑 ∧ 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) → 𝜒 ) ↔ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ∧ 𝜑 ) → 𝜒 ) )
4	1 3	mpbir	⊢ ( ( 𝜑 ∧ 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) → 𝜒 )
5		3anass	⊢ ( ( 𝜑 ∧ 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) ↔ ( 𝜑 ∧ ( 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) ) )
6		anabs5	⊢ ( ( 𝜑 ∧ ( 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) ) ↔ ( 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) )
7	5 6	bitri	⊢ ( ( 𝜑 ∧ 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) ↔ ( 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) )
8		ancom	⊢ ( ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜓 ∧ 𝜑 ) )
9	8	anbi2i	⊢ ( ( 𝜑 ∧ ( 𝜑 ∧ 𝜓 ) ) ↔ ( 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) )
10	7 9	bitr4i	⊢ ( ( 𝜑 ∧ 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) ↔ ( 𝜑 ∧ ( 𝜑 ∧ 𝜓 ) ) )
11		anabs5	⊢ ( ( 𝜑 ∧ ( 𝜑 ∧ 𝜓 ) ) ↔ ( 𝜑 ∧ 𝜓 ) )
12	11 8	bitri	⊢ ( ( 𝜑 ∧ ( 𝜑 ∧ 𝜓 ) ) ↔ ( 𝜓 ∧ 𝜑 ) )
13	10 12	bitri	⊢ ( ( 𝜑 ∧ 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) ↔ ( 𝜓 ∧ 𝜑 ) )
14	13	imbi1i	⊢ ( ( ( 𝜑 ∧ 𝜑 ∧ ( 𝜓 ∧ 𝜑 ) ) → 𝜒 ) ↔ ( ( 𝜓 ∧ 𝜑 ) → 𝜒 ) )
15	4 14	mpbi	⊢ ( ( 𝜓 ∧ 𝜑 ) → 𝜒 )