Theorem rnlem 931

Description: Lemma used in construction of real numbers. (Contributed by NM, 4-Sep-1995.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
rnlem	⊢ (((φ ∧ ψ) ∧ (χ ∧ θ)) ↔ (((φ ∧ χ) ∧ (ψ ∧ θ)) ∧ ((φ ∧ θ) ∧ (ψ ∧ χ))))

Proof of Theorem rnlem

Step	Hyp	Ref	Expression
1		an4 797	. . . 4 ⊢ (((φ ∧ ψ) ∧ (χ ∧ θ)) ↔ ((φ ∧ χ) ∧ (ψ ∧ θ)))
2	1	biimpi 186	. . 3 ⊢ (((φ ∧ ψ) ∧ (χ ∧ θ)) → ((φ ∧ χ) ∧ (ψ ∧ θ)))
3		an42 798	. . . 4 ⊢ (((φ ∧ θ) ∧ (ψ ∧ χ)) ↔ ((φ ∧ ψ) ∧ (χ ∧ θ)))
4	3	biimpri 197	. . 3 ⊢ (((φ ∧ ψ) ∧ (χ ∧ θ)) → ((φ ∧ θ) ∧ (ψ ∧ χ)))
5	2, 4	jca 518	. 2 ⊢ (((φ ∧ ψ) ∧ (χ ∧ θ)) → (((φ ∧ χ) ∧ (ψ ∧ θ)) ∧ ((φ ∧ θ) ∧ (ψ ∧ χ))))
6	3	biimpi 186	. . 3 ⊢ (((φ ∧ θ) ∧ (ψ ∧ χ)) → ((φ ∧ ψ) ∧ (χ ∧ θ)))
7	6	adantl 452	. 2 ⊢ ((((φ ∧ χ) ∧ (ψ ∧ θ)) ∧ ((φ ∧ θ) ∧ (ψ ∧ χ))) → ((φ ∧ ψ) ∧ (χ ∧ θ)))
8	5, 7	impbii 180	1 ⊢ (((φ ∧ ψ) ∧ (χ ∧ θ)) ↔ (((φ ∧ χ) ∧ (ψ ∧ θ)) ∧ ((φ ∧ θ) ∧ (ψ ∧ χ))))

Syntax hints:   ↔ wb 176   ∧ wa 358

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 177  df-an 360

This theorem is referenced by: (None)