Theorem rnlem 579

Description: Lemma used in construction of real numbers.

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		anandir 393	. 2 ⊢ (((φ ∧ ψ) ∧ (χ ∧ θ)) ↔ ((φ ∧ (χ ∧ θ)) ∧ (ψ ∧ (χ ∧ θ))))
2		anandi 392	. . 3 ⊢ ((φ ∧ (χ ∧ θ)) ↔ ((φ ∧ χ) ∧ (φ ∧ θ)))
3		anandi 392	. . 3 ⊢ ((ψ ∧ (χ ∧ θ)) ↔ ((ψ ∧ χ) ∧ (ψ ∧ θ)))
4	2, 3	anbi12i 369	. 2 ⊢ (((φ ∧ (χ ∧ θ)) ∧ (ψ ∧ (χ ∧ θ))) ↔ (((φ ∧ χ) ∧ (φ ∧ θ)) ∧ ((ψ ∧ χ) ∧ (ψ ∧ θ))))
5		ancom 333	. . . 4 ⊢ (((ψ ∧ χ) ∧ (ψ ∧ θ)) ↔ ((ψ ∧ θ) ∧ (ψ ∧ χ)))
6	5	anbi2i 367	. . 3 ⊢ ((((φ ∧ χ) ∧ (φ ∧ θ)) ∧ ((ψ ∧ χ) ∧ (ψ ∧ θ))) ↔ (((φ ∧ χ) ∧ (φ ∧ θ)) ∧ ((ψ ∧ θ) ∧ (ψ ∧ χ))))
7		an4 388	. . 3 ⊢ ((((φ ∧ χ) ∧ (φ ∧ θ)) ∧ ((ψ ∧ θ) ∧ (ψ ∧ χ))) ↔ (((φ ∧ χ) ∧ (ψ ∧ θ)) ∧ ((φ ∧ θ) ∧ (ψ ∧ χ))))
8	6, 7	bitr 151	. 2 ⊢ ((((φ ∧ χ) ∧ (φ ∧ θ)) ∧ ((ψ ∧ χ) ∧ (ψ ∧ θ))) ↔ (((φ ∧ χ) ∧ (ψ ∧ θ)) ∧ ((φ ∧ θ) ∧ (ψ ∧ χ))))
9	1, 4, 8	3bitr 155	1 ⊢ (((φ ∧ ψ) ∧ (χ ∧ θ)) ↔ (((φ ∧ χ) ∧ (ψ ∧ θ)) ∧ ((φ ∧ θ) ∧ (ψ ∧ χ))))

Syntax hints:   ↔ wb 127   ∧ wa 196

This theorem is referenced by:  mulcmpblnr 3977

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-an 198

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