Theorem 19.23t 1665

Description: Closed form of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)

Assertion

Ref	Expression
19.23t	⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)))

Proof of Theorem 19.23t

Step	Hyp	Ref	Expression
1		exim 1587	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
2		19.9t 1630	. . . 4 ⊢ (Ⅎ𝑥𝜓 → (∃𝑥𝜓 ↔ 𝜓))
3	2	biimpd 143	. . 3 ⊢ (Ⅎ𝑥𝜓 → (∃𝑥𝜓 → 𝜓))
4	1, 3	syl9r 73	. 2 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → 𝜓)))
5		nfr 1506	. . . 4 ⊢ (Ⅎ𝑥𝜓 → (𝜓 → ∀𝑥𝜓))
6	5	imim2d 54	. . 3 ⊢ (Ⅎ𝑥𝜓 → ((∃𝑥𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
7		19.38 1664	. . 3 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓))
8	6, 7	syl6 33	. 2 ⊢ (Ⅎ𝑥𝜓 → ((∃𝑥𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓)))
9	4, 8	impbid 128	1 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)))

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1341  Ⅎwnf 1448  ∃wex 1480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-nf 1449

This theorem is referenced by:  19.23  1666  r19.23t  2573  ceqsalt  2752  vtoclgft  2776  sbciegft  2981