Theorem r19.26-3 3213

Description: Version of r19.26 3211 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 1072	. . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒))
2	1	ralbii 3128	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ∀𝑥 ∈ 𝐴 ((𝜑 ∧ 𝜓) ∧ 𝜒))
3		r19.26 3211	. 2 ⊢ (∀𝑥 ∈ 𝐴 ((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ ∀𝑥 ∈ 𝐴 𝜒))
4		r19.26 3211	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓))
5	4	anbi1i 602	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ ∀𝑥 ∈ 𝐴 𝜒) ↔ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) ∧ ∀𝑥 ∈ 𝐴 𝜒))
6		df-3an 1072	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓 ∧ ∀𝑥 ∈ 𝐴 𝜒) ↔ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) ∧ ∀𝑥 ∈ 𝐴 𝜒))
7	5, 6	bitr4i 267	. 2 ⊢ ((∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ ∀𝑥 ∈ 𝐴 𝜒) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓 ∧ ∀𝑥 ∈ 𝐴 𝜒))
8	2, 3, 7	3bitri 286	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓 ∧ ∀𝑥 ∈ 𝐴 𝜒))

Syntax hints:   ↔ wb 196   ∧ wa 382   ∧ w3a 1070  ∀wral 3060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884

This theorem depends on definitions:  df-bi 197  df-an 383  df-3an 1072  df-ral 3065

This theorem is referenced by:  sgrp2rid2ex  17621  axeuclid  26063  axcontlem8  26071  stoweidlem60  40788