Theorem 19.41rg

Description: Closed form of right-to-left implication of 19.41 , Theorem 19.41 of Margaris p. 90. Derived from 19.41rgVD . (Contributed by Alan Sare, 8-Feb-2014) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	19.41rg	⊢ ( ∀ 𝑥 ( 𝜓 → ∀ 𝑥 𝜓 ) → ( ( ∃ 𝑥 𝜑 ∧ 𝜓 ) → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		sp	⊢ ( ∀ 𝑥 ( 𝜓 → ∀ 𝑥 𝜓 ) → ( 𝜓 → ∀ 𝑥 𝜓 ) )
2		pm3.21	⊢ ( 𝜓 → ( 𝜑 → ( 𝜑 ∧ 𝜓 ) ) )
3	2	a1i	⊢ ( ( 𝜓 → ∀ 𝑥 𝜓 ) → ( 𝜓 → ( 𝜑 → ( 𝜑 ∧ 𝜓 ) ) ) )
4	3	al2imi	⊢ ( ∀ 𝑥 ( 𝜓 → ∀ 𝑥 𝜓 ) → ( ∀ 𝑥 𝜓 → ∀ 𝑥 ( 𝜑 → ( 𝜑 ∧ 𝜓 ) ) ) )
5		exim	⊢ ( ∀ 𝑥 ( 𝜑 → ( 𝜑 ∧ 𝜓 ) ) → ( ∃ 𝑥 𝜑 → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) )
6	4 5	syl6	⊢ ( ∀ 𝑥 ( 𝜓 → ∀ 𝑥 𝜓 ) → ( ∀ 𝑥 𝜓 → ( ∃ 𝑥 𝜑 → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) ) )
7	1 6	syld	⊢ ( ∀ 𝑥 ( 𝜓 → ∀ 𝑥 𝜓 ) → ( 𝜓 → ( ∃ 𝑥 𝜑 → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) ) )
8	7	com23	⊢ ( ∀ 𝑥 ( 𝜓 → ∀ 𝑥 𝜓 ) → ( ∃ 𝑥 𝜑 → ( 𝜓 → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) ) )
9	8	impd	⊢ ( ∀ 𝑥 ( 𝜓 → ∀ 𝑥 𝜓 ) → ( ( ∃ 𝑥 𝜑 ∧ 𝜓 ) → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) )