Theorem r19.29uz

Description: A version of 19.29 for upper integer quantifiers. (Contributed by Mario Carneiro, 10-Feb-2014)

	Ref	Expression
Hypothesis	rexuz3.1	$$⊢Z={\mathbb{Z}}_{\ge M}$$
Assertion	r19.29uz	$$⊢\left(\forall k\in Z\phi \wedge \exists j\in Z\forall k\in {\mathbb{Z}}_{\ge j}\psi \right)\to \exists j\in Z\forall k\in {\mathbb{Z}}_{\ge j}\left(\phi \wedge \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		rexuz3.1	$$⊢Z={\mathbb{Z}}_{\ge M}$$
2	1	uztrn2	$$⊢\left(j\in Z\wedge k\in {\mathbb{Z}}_{\ge j}\right)\to k\in Z$$
3	2	ex	$$⊢j\in Z\to \left(k\in {\mathbb{Z}}_{\ge j}\to k\in Z\right)$$
4		pm3.2	$$⊢\phi \to \left(\psi \to \left(\phi \wedge \psi \right)\right)$$
5	4	a1i	$$⊢j\in Z\to \left(\phi \to \left(\psi \to \left(\phi \wedge \psi \right)\right)\right)$$
6	3 5	imim12d	$$⊢j\in Z\to \left(\left(k\in Z\to \phi \right)\to \left(k\in {\mathbb{Z}}_{\ge j}\to \left(\psi \to \left(\phi \wedge \psi \right)\right)\right)\right)$$
7	6	ralimdv2	$$⊢j\in Z\to \left(\forall k\in Z\phi \to \forall k\in {\mathbb{Z}}_{\ge j}\left(\psi \to \left(\phi \wedge \psi \right)\right)\right)$$
8	7	impcom	$$⊢\left(\forall k\in Z\phi \wedge j\in Z\right)\to \forall k\in {\mathbb{Z}}_{\ge j}\left(\psi \to \left(\phi \wedge \psi \right)\right)$$
9		ralim	$$⊢\forall k\in {\mathbb{Z}}_{\ge j}\left(\psi \to \left(\phi \wedge \psi \right)\right)\to \left(\forall k\in {\mathbb{Z}}_{\ge j}\psi \to \forall k\in {\mathbb{Z}}_{\ge j}\left(\phi \wedge \psi \right)\right)$$
10	8 9	syl	$$⊢\left(\forall k\in Z\phi \wedge j\in Z\right)\to \left(\forall k\in {\mathbb{Z}}_{\ge j}\psi \to \forall k\in {\mathbb{Z}}_{\ge j}\left(\phi \wedge \psi \right)\right)$$
11	10	reximdva	$$⊢\forall k\in Z\phi \to \left(\exists j\in Z\forall k\in {\mathbb{Z}}_{\ge j}\psi \to \exists j\in Z\forall k\in {\mathbb{Z}}_{\ge j}\left(\phi \wedge \psi \right)\right)$$
12	11	imp	$$⊢\left(\forall k\in Z\phi \wedge \exists j\in Z\forall k\in {\mathbb{Z}}_{\ge j}\psi \right)\to \exists j\in Z\forall k\in {\mathbb{Z}}_{\ge j}\left(\phi \wedge \psi \right)$$