Theorem uun2221p1 42323

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
uun2221p1.1	⊢ ((𝜑 ∧ (𝜓 ∧ 𝜑) ∧ 𝜑) → 𝜒)

Assertion

Ref	Expression
uun2221p1	⊢ ((𝜓 ∧ 𝜑) → 𝜒)

Proof of Theorem uun2221p1

Step	Hyp	Ref	Expression
1		uun2221p1.1	. . 3 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜑) ∧ 𝜑) → 𝜒)
2		3anrot 1098	. . . 4 ⊢ ((𝜑 ∧ 𝜑 ∧ (𝜓 ∧ 𝜑)) ↔ (𝜑 ∧ (𝜓 ∧ 𝜑) ∧ 𝜑))
3	2	imbi1i 349	. . 3 ⊢ (((𝜑 ∧ 𝜑 ∧ (𝜓 ∧ 𝜑)) → 𝜒) ↔ ((𝜑 ∧ (𝜓 ∧ 𝜑) ∧ 𝜑) → 𝜒))
4	1, 3	mpbir 230	. 2 ⊢ ((𝜑 ∧ 𝜑 ∧ (𝜓 ∧ 𝜑)) → 𝜒)
5		3anass 1093	. . . . . 6 ⊢ ((𝜑 ∧ 𝜑 ∧ (𝜓 ∧ 𝜑)) ↔ (𝜑 ∧ (𝜑 ∧ (𝜓 ∧ 𝜑))))
6		anabs5 659	. . . . . 6 ⊢ ((𝜑 ∧ (𝜑 ∧ (𝜓 ∧ 𝜑))) ↔ (𝜑 ∧ (𝜓 ∧ 𝜑)))
7	5, 6	bitri 274	. . . . 5 ⊢ ((𝜑 ∧ 𝜑 ∧ (𝜓 ∧ 𝜑)) ↔ (𝜑 ∧ (𝜓 ∧ 𝜑)))
8		ancom 460	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑))
9	8	anbi2i 622	. . . . 5 ⊢ ((𝜑 ∧ (𝜑 ∧ 𝜓)) ↔ (𝜑 ∧ (𝜓 ∧ 𝜑)))
10	7, 9	bitr4i 277	. . . 4 ⊢ ((𝜑 ∧ 𝜑 ∧ (𝜓 ∧ 𝜑)) ↔ (𝜑 ∧ (𝜑 ∧ 𝜓)))
11		anabs5 659	. . . . 5 ⊢ ((𝜑 ∧ (𝜑 ∧ 𝜓)) ↔ (𝜑 ∧ 𝜓))
12	11, 8	bitri 274	. . . 4 ⊢ ((𝜑 ∧ (𝜑 ∧ 𝜓)) ↔ (𝜓 ∧ 𝜑))
13	10, 12	bitri 274	. . 3 ⊢ ((𝜑 ∧ 𝜑 ∧ (𝜓 ∧ 𝜑)) ↔ (𝜓 ∧ 𝜑))
14	13	imbi1i 349	. 2 ⊢ (((𝜑 ∧ 𝜑 ∧ (𝜓 ∧ 𝜑)) → 𝜒) ↔ ((𝜓 ∧ 𝜑) → 𝜒))
15	4, 14	mpbi 229	1 ⊢ ((𝜓 ∧ 𝜑) → 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: (None)