Theorem uun0.1 42398

Description: Convention notation form of un0.1 42399. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
uun0.1.1	⊢ (⊤ → 𝜑)
uun0.1.2	⊢ (𝜓 → 𝜒)
uun0.1.3	⊢ ((⊤ ∧ 𝜓) → 𝜃)

Assertion

Ref	Expression
uun0.1	⊢ (𝜓 → 𝜃)

Proof of Theorem uun0.1

Step	Hyp	Ref	Expression
1		tru 1543	. 2 ⊢ ⊤
2		uun0.1.1	. . . . . 6 ⊢ (⊤ → 𝜑)
3		uun0.1.2	. . . . . 6 ⊢ (𝜓 → 𝜒)
4	2, 3	pm3.2i 471	. . . . 5 ⊢ ((⊤ → 𝜑) ∧ (𝜓 → 𝜒))
5		uun0.1.3	. . . . 5 ⊢ ((⊤ ∧ 𝜓) → 𝜃)
6	4, 5	pm3.2i 471	. . . 4 ⊢ (((⊤ → 𝜑) ∧ (𝜓 → 𝜒)) ∧ ((⊤ ∧ 𝜓) → 𝜃))
7	6	simpri 486	. . 3 ⊢ ((⊤ ∧ 𝜓) → 𝜃)
8	7	ex 413	. 2 ⊢ (⊤ → (𝜓 → 𝜃))
9	1, 8	ax-mp 5	1 ⊢ (𝜓 → 𝜃)

Syntax hints:   → wi 4   ∧ wa 396  ⊤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-tru 1542

This theorem is referenced by:  un0.1  42399  sspwimp  42538