Theorem 19.26-3an 1871

Description: Theorem 19.26 1869 with triple conjunction. (Contributed by NM, 13-Sep-2011.)

Assertion

Ref	Expression
19.26-3an	⊢ (∀𝑥(𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓 ∧ ∀𝑥𝜒))

Proof of Theorem 19.26-3an

Step	Hyp	Ref	Expression
1		19.26 1869	. . 3 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))
2	1	anbi1i 623	. 2 ⊢ ((∀𝑥(𝜑 ∧ 𝜓) ∧ ∀𝑥𝜒) ↔ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) ∧ ∀𝑥𝜒))
3		df-3an 1089	. . . 4 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒))
4	3	albii 1817	. . 3 ⊢ (∀𝑥(𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ∀𝑥((𝜑 ∧ 𝜓) ∧ 𝜒))
5		19.26 1869	. . 3 ⊢ (∀𝑥((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ (∀𝑥(𝜑 ∧ 𝜓) ∧ ∀𝑥𝜒))
6	4, 5	bitri 275	. 2 ⊢ (∀𝑥(𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (∀𝑥(𝜑 ∧ 𝜓) ∧ ∀𝑥𝜒))
7		df-3an 1089	. 2 ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓 ∧ ∀𝑥𝜒) ↔ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) ∧ ∀𝑥𝜒))
8	2, 6, 7	3bitr4i 303	1 ⊢ (∀𝑥(𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓 ∧ ∀𝑥𝜒))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1087  ∀wal 1535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089

This theorem is referenced by:  alrim3con13v  44504  19.21a3con13vVD  44823