Theorem r19.26-3 2681

Description: Theorem 19.26 of [Margaris] p. 90 with 3 restricted 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 1011	. . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒))
2	1	ralbii 2556	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ∀𝑥 ∈ 𝐴 ((𝜑 ∧ 𝜓) ∧ 𝜒))
3		r19.26 2677	. 2 ⊢ (∀𝑥 ∈ 𝐴 ((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ ∀𝑥 ∈ 𝐴 𝜒))
4		r19.26 2677	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓))
5	4	anbi1i 462	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ ∀𝑥 ∈ 𝐴 𝜒) ↔ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) ∧ ∀𝑥 ∈ 𝐴 𝜒))
6		df-3an 1011	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓 ∧ ∀𝑥 ∈ 𝐴 𝜒) ↔ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) ∧ ∀𝑥 ∈ 𝐴 𝜒))
7	5, 6	bitr4i 187	. 2 ⊢ ((∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ ∀𝑥 ∈ 𝐴 𝜒) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓 ∧ ∀𝑥 ∈ 𝐴 𝜒))
8	2, 3, 7	3bitri 206	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓 ∧ ∀𝑥 ∈ 𝐴 𝜒))

This proof depends on syntax axioms:   ∧ wa 104   ↔ wb 105   ∧ w3a 1009  ∀wral 2528

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-4 1563  ax-17 1579

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-ral 2533

This theorem is used by: (None)