Theorem aleximi 1908

Description: A variant of al2imi 1892: instead of applying ∀𝑥 quantifiers to the final implication, replace them with ∃𝑥. A shorter proof is possible using nfa1 2177, sps 2202 and eximd 2232, but it depends on more axioms. (Contributed by Wolf Lammen, 18-Aug-2019.)

Hypothesis

Ref	Expression
aleximi.1	⊢ (𝜑 → (𝜓 → 𝜒))

Assertion

Ref	Expression
aleximi	⊢ (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))

Proof of Theorem aleximi

Step	Hyp	Ref	Expression
1		aleximi.1	. . . . 5 ⊢ (𝜑 → (𝜓 → 𝜒))
2	1	con3d 148	. . . 4 ⊢ (𝜑 → (¬ 𝜒 → ¬ 𝜓))
3	2	al2imi 1892	. . 3 ⊢ (∀𝑥𝜑 → (∀𝑥 ¬ 𝜒 → ∀𝑥 ¬ 𝜓))
4		alnex 1855	. . 3 ⊢ (∀𝑥 ¬ 𝜒 ↔ ¬ ∃𝑥𝜒)
5		alnex 1855	. . 3 ⊢ (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓)
6	3, 4, 5	3imtr3g 284	. 2 ⊢ (∀𝑥𝜑 → (¬ ∃𝑥𝜒 → ¬ ∃𝑥𝜓))
7	6	con4d 114	1 ⊢ (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1630  ∃wex 1853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886

This theorem depends on definitions:  df-bi 197  df-ex 1854

This theorem is referenced by:  alexbii  1909  exim  1910  exanOLD  1938  eximdh  1940  19.29  1950  19.29r  1951  19.35  1954  19.25  1957  19.30  1958  19.40b  1964  speimfw  2042  aeveq  2133  2ax6elem  2586  mo3  2645  mopick  2673  2mo  2689  ssopab2  5151  ssoprab2  6876  axextnd  9605  bj-2exim  32901  bj-exalimi  32918  bj-sb56  32945  wl-dveeq12  33624  wl-mo3t  33671  pm10.56  39071  2exim  39080