Theorem 19.28v 2026

Description: Version of 19.28 2264 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 1900	. 2 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))
2		19.3v 2012	. 2 ⊢ (∀𝑥𝜑 ↔ 𝜑)
3	1, 2	bianbi 638	1 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 400  ∀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  ax-6 1997

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1810

This theorem is used by:  reu6  3689  dfer2  8691  kmlem14  10152  kmlem15  10153  bnj1176  35402  bnj1186  35404  mh-infprim2bi  37086  ismnuprim  45032  19.28vv  45124