Metamath Proof Explorer

Theorem mdandyvrx4

Description: Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016)

		Ref	Expression
	Hypotheses	mdandyvrx4.1	$⊢ (φ ⊻ ζ)$
		mdandyvrx4.2	$⊢ (ψ ⊻ σ)$
		mdandyvrx4.3	$⊢ χ \leftrightarrow φ$
		mdandyvrx4.4	$⊢ θ \leftrightarrow φ$
		mdandyvrx4.5	$⊢ τ \leftrightarrow ψ$
		mdandyvrx4.6	$⊢ η \leftrightarrow φ$
	Assertion	mdandyvrx4	$⊢ ((((χ ⊻ ζ) \land (θ ⊻ ζ)) \land (τ ⊻ σ)) \land (η ⊻ ζ))$

Proof

Step	Hyp	Ref	Expression
1		mdandyvrx4.1	$⊢ (φ ⊻ ζ)$
2		mdandyvrx4.2	$⊢ (ψ ⊻ σ)$
3		mdandyvrx4.3	$⊢ χ \leftrightarrow φ$
4		mdandyvrx4.4	$⊢ θ \leftrightarrow φ$
5		mdandyvrx4.5	$⊢ τ \leftrightarrow ψ$
6		mdandyvrx4.6	$⊢ η \leftrightarrow φ$
7	1 3	axorbciffatcxorb	$⊢ (χ ⊻ ζ)$
8	1 4	axorbciffatcxorb	$⊢ (θ ⊻ ζ)$
9	7 8	pm3.2i	$⊢ ((χ ⊻ ζ) \land (θ ⊻ ζ))$
10	2 5	axorbciffatcxorb	$⊢ (τ ⊻ σ)$
11	9 10	pm3.2i	$⊢ (((χ ⊻ ζ) \land (θ ⊻ ζ)) \land (τ ⊻ σ))$
12	1 6	axorbciffatcxorb	$⊢ (η ⊻ ζ)$
13	11 12	pm3.2i	$⊢ ((((χ ⊻ ζ) \land (θ ⊻ ζ)) \land (τ ⊻ σ)) \land (η ⊻ ζ))$