Theorem exim 1610

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 1551	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → ∀𝑥∀𝑥(𝜑 → 𝜓))
2		hbe1 1506	. 2 ⊢ (∃𝑥𝜓 → ∀𝑥∃𝑥𝜓)
3		19.8a 1601	. . . 4 ⊢ (𝜓 → ∃𝑥𝜓)
4	3	imim2i 12	. . 3 ⊢ ((𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓))
5	4	sps 1548	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓))
6	1, 2, 5	exlimdh 1607	1 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))

Syntax hints:   → wi 4  ∀wal 1362  ∃wex 1503

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-ial 1545

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  eximi  1611  exbi  1615  eximdh  1622  19.29  1631  19.25  1637  alexim  1656  19.23t  1688  spimt  1747  equvini  1769  nfexd  1772  ax10oe  1808  sbcof2  1821  spsbim  1854  nf5-1  2036  mor  2080  rexim  2584  elex22  2767  elex2  2768  vtoclegft  2824  spcimgft  2828  spcimegft  2830  spc2gv  2843  spc3gv  2845  ssoprab2  5953  bj-inf2vnlem1  15200