Theorem un01 42409

Description: A unionizing deduction. (Contributed by Alan Sare, 28-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
un01.1	⊢ (   (   ⊤   ,   𝜑   )   ▶   𝜓   )

Assertion

Ref	Expression
un01	⊢ (   𝜑   ▶   𝜓   )

Proof of Theorem un01

Step	Hyp	Ref	Expression
1		tru 1543	. . . 4 ⊢ ⊤
2	1	jctl 524	. . 3 ⊢ (𝜑 → (⊤ ∧ 𝜑))
3		un01.1	. . . 4 ⊢ (   (   ⊤   ,   𝜑   )   ▶   𝜓   )
4	3	dfvd2ani 42203	. . 3 ⊢ ((⊤ ∧ 𝜑) → 𝜓)
5	2, 4	syl 17	. 2 ⊢ (𝜑 → 𝜓)
6	5	dfvd1ir 42193	1 ⊢ (   𝜑   ▶   𝜓   )

Syntax hints:   ∧ wa 396  ⊤wtru 1540  (   wvd1 42189  (   wvhc2 42200

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  df-vd1 42190  df-vhc2 42201

This theorem is referenced by: (None)