Theorem r19.26-3 3111

Description: Version of r19.26 3110 with three quantifiers. (Contributed by FL, 22-Nov-2010.)

Assertion

Ref	Expression
r19.26-3	⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓 ∧ ∀𝑥 ∈ 𝐴 𝜒))

Proof of Theorem r19.26-3

Step	Hyp	Ref	Expression
1		r19.26 3110	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ ∀𝑥 ∈ 𝐴 𝜒))
2		r19.26 3110	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓))
3	1, 2	bianbi 627	. 2 ⊢ (∀𝑥 ∈ 𝐴 ((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) ∧ ∀𝑥 ∈ 𝐴 𝜒))
4		df-3an 1088	. . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒))
5	4	ralbii 3092	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ∀𝑥 ∈ 𝐴 ((𝜑 ∧ 𝜓) ∧ 𝜒))
6		df-3an 1088	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓 ∧ ∀𝑥 ∈ 𝐴 𝜒) ↔ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) ∧ ∀𝑥 ∈ 𝐴 𝜒))
7	3, 5, 6	3bitr4i 303	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓 ∧ ∀𝑥 ∈ 𝐴 𝜒))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∀wral 3060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-ral 3061

This theorem is referenced by:  sgrp2rid2ex  18941  axeuclid  28979  axcontlem8  28987  stoweidlem60  46080