Theorem uunT1 42400

Description: A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 3-Dec-2015.) Proof was revised to accommodate a possible future version of df-tru 1542. (Revised by David A. Wheeler, 8-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
uunT1.1	⊢ ((⊤ ∧ 𝜑) → 𝜓)

Assertion

Ref	Expression
uunT1	⊢ (𝜑 → 𝜓)

Proof of Theorem uunT1

Step	Hyp	Ref	Expression
1		orc 864	. . 3 ⊢ (𝜑 → (𝜑 ∨ ¬ 𝜑))
2		tru 1543	. . . . 5 ⊢ ⊤
3		biid 260	. . . . 5 ⊢ (𝜑 ↔ 𝜑)
4	2, 3	2th 263	. . . 4 ⊢ (⊤ ↔ (𝜑 ↔ 𝜑))
5		exmid 892	. . . . . 6 ⊢ (𝜑 ∨ ¬ 𝜑)
6	5	a1i 11	. . . . 5 ⊢ ((𝜑 ↔ 𝜑) → (𝜑 ∨ ¬ 𝜑))
7		biidd 261	. . . . 5 ⊢ ((𝜑 ∨ ¬ 𝜑) → (𝜑 ↔ 𝜑))
8	6, 7	impbii 208	. . . 4 ⊢ ((𝜑 ↔ 𝜑) ↔ (𝜑 ∨ ¬ 𝜑))
9	4, 8	bitri 274	. . 3 ⊢ (⊤ ↔ (𝜑 ∨ ¬ 𝜑))
10	1, 9	sylibr 233	. 2 ⊢ (𝜑 → ⊤)
11		uunT1.1	. 2 ⊢ ((⊤ ∧ 𝜑) → 𝜓)
12	10, 11	mpancom 685	1 ⊢ (𝜑 → 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844  ⊤wtru 1540

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 397  df-or 845  df-tru 1542

This theorem is referenced by:  uunT21  42402  sspwimpALT  42545