Theorem 19.28v 2030

Description: Version of 19.28 2267 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 1902	. 2 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))
2		19.3v 2016	. 2 ⊢ (∀𝑥𝜑 ↔ 𝜑)
3	1, 2	bianbi 639	1 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401  ∀wal 1567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812

This theorem is used by:  reu6  3687  dfer2  8732  kmlem14  10255  kmlem15  10256  bnj1176  35691  bnj1186  35693  reu6f  35765  onprcf1acwevdlem1  36002  mh-infprim2bi  37468  ismnuprim  45424  19.28vv  45516