Theorem 19.28v 2039

Description: Version of 19.28 2213 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 25-Mar-2004.)

Assertion

Ref	Expression
19.28v	⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))

Distinct variable group:   𝜑,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem 19.28v

Step	Hyp	Ref	Expression
1		19.26 1916	. 2 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))
2		19.3v 2031	. . 3 ⊢ (∀𝑥𝜑 ↔ 𝜑)
3	2	anbi1i 617	. 2 ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))
4	1, 3	bitri 267	1 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))

Syntax hints:   ↔ wb 198   ∧ wa 386  ∀wal 1599

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021

This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1824

This theorem is referenced by:  reu6  3606  dfer2  8027  kmlem14  9320  kmlem15  9321  bnj1176  31686  bnj1186  31688  19.28vv  39533