Theorem 19.28v 2018

Description: Version of 19.28 2265 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 1892	. 2 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))
2		19.3v 2004	. 2 ⊢ (∀𝑥𝜑 ↔ 𝜑)
3	1, 2	bianbi 636	1 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))

Syntax hints:   ↔ wb 208   ∧ wa 399  ∀wal 1560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1802

This theorem is referenced by:  reu6  3691  dfer2  8681  kmlem14  10122  kmlem15  10123  bnj1176  35302  bnj1186  35304  mh-infprim2bi  36912  ismnuprim  44875  19.28vv  44967