Theorem 19.27v 2024

Description: Version of 19.27 2262 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 3-Jun-2004.)

Assertion

Ref	Expression
19.27v	⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ 𝜓))

Distinct variable group:   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem 19.27v

Step	Hyp	Ref	Expression
1		19.26 1899	. 2 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))
2		19.3v 2011	. . 3 ⊢ (∀𝑥𝜓 ↔ 𝜓)
3	2	anbi2i 634	. 2 ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) ↔ (∀𝑥𝜑 ∧ 𝜓))
4	1, 3	bitri 278	1 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ 𝜓))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 400  ∀wal 1567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809

This theorem is used by:  rexrsb  47865