Theorem r19.26-3 3097

Description: Version of r19.26 3095 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		df-3an 1088	. . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒))
2	1	ralbii 3092	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ∀𝑥 ∈ 𝐴 ((𝜑 ∧ 𝜓) ∧ 𝜒))
3		r19.26 3095	. 2 ⊢ (∀𝑥 ∈ 𝐴 ((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ ∀𝑥 ∈ 𝐴 𝜒))
4		r19.26 3095	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓))
5	4	anbi1i 624	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ ∀𝑥 ∈ 𝐴 𝜒) ↔ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) ∧ ∀𝑥 ∈ 𝐴 𝜒))
6		df-3an 1088	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓 ∧ ∀𝑥 ∈ 𝐴 𝜒) ↔ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) ∧ ∀𝑥 ∈ 𝐴 𝜒))
7	5, 6	bitr4i 277	. 2 ⊢ ((∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ ∀𝑥 ∈ 𝐴 𝜒) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓 ∧ ∀𝑥 ∈ 𝐴 𝜒))
8	2, 3, 7	3bitri 297	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓 ∧ ∀𝑥 ∈ 𝐴 𝜒))

Syntax hints:   ↔ wb 205   ∧ wa 396   ∧ w3a 1086  ∀wral 3064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812

This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1088  df-ral 3069

This theorem is referenced by:  sgrp2rid2ex  18566  axeuclid  27331  axcontlem8  27339  stoweidlem60  43601