Theorem 19.38 1699

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 1518	. . 3 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)
2		hba1 1563	. . 3 ⊢ (∀𝑥𝜓 → ∀𝑥∀𝑥𝜓)
3	1, 2	hbim 1568	. 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → ∀𝑥𝜓))
4		19.8a 1613	. . 3 ⊢ (𝜑 → ∃𝑥𝜑)
5		ax-4 1533	. . 3 ⊢ (∀𝑥𝜓 → 𝜓)
6	4, 5	imim12i 59	. 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → (𝜑 → 𝜓))
7	3, 6	alrimih 1492	1 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓))

Syntax hints:   → wi 4  ∀wal 1371  ∃wex 1515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533  ax-ial 1557  ax-i5r 1558

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  19.23t  1700  sbi2v  1916  mo3h  2107  ralm  3564