Theorem 19.28v 1997

Description: Version of 19.28 2231 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 1871	. 2 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))
2		19.3v 1983	. 2 ⊢ (∀𝑥𝜑 ↔ 𝜑)
3	1, 2	bianbi 627	1 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))

Syntax hints:   ↔ wb 206   ∧ wa 395  ∀wal 1539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781

This theorem is referenced by:  reu6  3680  dfer2  8623  kmlem14  10055  kmlem15  10056  bnj1176  35017  bnj1186  35019  ismnuprim  44386  19.28vv  44478