Theorem 19.32v 1970

Description: Version of 19.32 2269 with a disjoint variable condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.)

Assertion

Ref	Expression
19.32v	⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓))

Distinct variable group:   𝜑,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem 19.32v

Step	Hyp	Ref	Expression
1		19.21v 1969	. 2 ⊢ (∀𝑥(¬ 𝜑 → 𝜓) ↔ (¬ 𝜑 → ∀𝑥𝜓))
2		df-or 861	. . 3 ⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓))
3	2	albii 1849	. 2 ⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ ∀𝑥(¬ 𝜑 → 𝜓))
4		df-or 861	. 2 ⊢ ((𝜑 ∨ ∀𝑥𝜓) ↔ (¬ 𝜑 → ∀𝑥𝜓))
5	1, 3, 4	3bitr4i 306	1 ⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∨ wo 860  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940

This proof depends on definitions:  df-bi 210  df-or 861  df-ex 1810

This theorem is used by:  19.31v  1971  iresn0n0  6056  pm10.12  45096