Theorem exim 1648

Description: Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.)

Assertion

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

Proof of Theorem exim

Step	Hyp	Ref	Expression
1		hba1 1589	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → ∀𝑥∀𝑥(𝜑 → 𝜓))
2		hbe1 1544	. 2 ⊢ (∃𝑥𝜓 → ∀𝑥∃𝑥𝜓)
3		19.8a 1639	. . . 4 ⊢ (𝜓 → ∃𝑥𝜓)
4	3	imim2i 12	. . 3 ⊢ ((𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓))
5	4	sps 1586	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓))
6	1, 2, 5	exlimdh 1645	1 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))

Syntax hints:   → wi 4  ∀wal 1396  ∃wex 1541

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-ial 1583

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  eximi  1649  exbi  1653  eximdh  1660  19.29  1669  19.25  1675  alexim  1694  19.23t  1725  spimt  1784  equvini  1806  nfexd  1809  ax10oe  1845  sbcof2  1858  spsbim  1891  nf5-1  2077  mor  2122  rexim  2627  elex22  2819  elex2  2820  vtoclegft  2879  spcimgft  2883  spcimegft  2885  spc2gv  2898  spc3gv  2900  ssoprab2  6087  bj-inf2vnlem1  16666