Theorem exim 1578

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 1520	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → ∀𝑥∀𝑥(𝜑 → 𝜓))
2		hbe1 1471	. 2 ⊢ (∃𝑥𝜓 → ∀𝑥∃𝑥𝜓)
3		19.8a 1569	. . . 4 ⊢ (𝜓 → ∃𝑥𝜓)
4	3	imim2i 12	. . 3 ⊢ ((𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓))
5	4	sps 1517	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓))
6	1, 2, 5	exlimdh 1575	1 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))

Syntax hints:   → wi 4  ∀wal 1329  ∃wex 1468

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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-ial 1514

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  eximi  1579  exbi  1583  eximdh  1590  19.29  1599  19.25  1605  alexim  1624  19.23t  1655  spimt  1714  equvini  1731  nfexd  1734  ax10oe  1769  sbcof2  1782  spsbim  1815  mor  2039  rexim  2524  elex22  2696  elex2  2697  vtoclegft  2753  spcimgft  2757  spcimegft  2759  spc2gv  2771  spc3gv  2773  ssoprab2  5820  bj-inf2vnlem1  13157