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Metamath Proof Explorer

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 ) ) )