Theorem 19.38 1606

Description: Theorem 19.38 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
19.38	⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓))

Proof of Theorem 19.38

Step	Hyp	Ref	Expression
1		hbe1 1424	. . 3 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)
2		hba1 1473	. . 3 ⊢ (∀𝑥𝜓 → ∀𝑥∀𝑥𝜓)
3	1, 2	hbim 1477	. 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → ∀𝑥𝜓))
4		19.8a 1522	. . 3 ⊢ (𝜑 → ∃𝑥𝜑)
5		ax-4 1440	. . 3 ⊢ (∀𝑥𝜓 → 𝜓)
6	4, 5	imim12i 58	. 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → (𝜑 → 𝜓))
7	3, 6	alrimih 1398	1 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓))

Syntax hints:   → wi 4  ∀wal 1282  ∃wex 1421

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  19.23t  1607  sbi2v  1813  mo3h  1994  rgenm  3343  ralm  3345