Theorem bnj887 32040

Description: ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj887.1	⊢ (𝜑 ↔ 𝜑′)
bnj887.2	⊢ (𝜓 ↔ 𝜓′)
bnj887.3	⊢ (𝜒 ↔ 𝜒′)
bnj887.4	⊢ (𝜃 ↔ 𝜃′)

Assertion

Ref	Expression
bnj887	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑′ ∧ 𝜓′ ∧ 𝜒′ ∧ 𝜃′))

Proof of Theorem bnj887

Step	Hyp	Ref	Expression
1		bnj887.1	. . . 4 ⊢ (𝜑 ↔ 𝜑′)
2		bnj887.2	. . . 4 ⊢ (𝜓 ↔ 𝜓′)
3		bnj887.3	. . . 4 ⊢ (𝜒 ↔ 𝜒′)
4	1, 2, 3	3anbi123i 1151	. . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜑′ ∧ 𝜓′ ∧ 𝜒′))
5		bnj887.4	. . 3 ⊢ (𝜃 ↔ 𝜃′)
6	4, 5	anbi12i 628	. 2 ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) ↔ ((𝜑′ ∧ 𝜓′ ∧ 𝜒′) ∧ 𝜃′))
7		df-bnj17 31961	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃))
8		df-bnj17 31961	. 2 ⊢ ((𝜑′ ∧ 𝜓′ ∧ 𝜒′ ∧ 𝜃′) ↔ ((𝜑′ ∧ 𝜓′ ∧ 𝜒′) ∧ 𝜃′))
9	6, 7, 8	3bitr4i 305	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑′ ∧ 𝜓′ ∧ 𝜒′ ∧ 𝜃′))

Syntax hints:   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   ∧ w-bnj17 31960

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

This theorem depends on definitions:  df-bi 209  df-an 399  df-3an 1085  df-bnj17 31961

This theorem is referenced by:  bnj1040  32248  bnj1128  32266