Theorem exim 1623

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 1564	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → ∀𝑥∀𝑥(𝜑 → 𝜓))
2		hbe1 1519	. 2 ⊢ (∃𝑥𝜓 → ∀𝑥∃𝑥𝜓)
3		19.8a 1614	. . . 4 ⊢ (𝜓 → ∃𝑥𝜓)
4	3	imim2i 12	. . 3 ⊢ ((𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓))
5	4	sps 1561	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓))
6	1, 2, 5	exlimdh 1620	1 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))

Syntax hints:   → wi 4  ∀wal 1371  ∃wex 1516

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 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-ial 1558

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  eximi  1624  exbi  1628  eximdh  1635  19.29  1644  19.25  1650  alexim  1669  19.23t  1701  spimt  1760  equvini  1782  nfexd  1785  ax10oe  1821  sbcof2  1834  spsbim  1867  nf5-1  2053  mor  2097  rexim  2601  elex22  2789  elex2  2790  vtoclegft  2849  spcimgft  2853  spcimegft  2855  spc2gv  2868  spc3gv  2870  ssoprab2  6013  bj-inf2vnlem1  16040