Theorem aleximi 1834

Description: A variant of al2imi 1817: instead of applying ∀𝑥 quantifiers to the final implication, replace them with ∃𝑥. A shorter proof is possible using nfa1 2148, sps 2178 and eximd 2209, 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 152	. . . 4 ⊢ (𝜑 → (¬ 𝜒 → ¬ 𝜓))
3	2	al2imi 1817	. . 3 ⊢ (∀𝑥𝜑 → (∀𝑥 ¬ 𝜒 → ∀𝑥 ¬ 𝜓))
4		alnex 1783	. . 3 ⊢ (∀𝑥 ¬ 𝜒 ↔ ¬ ∃𝑥𝜒)
5		alnex 1783	. . 3 ⊢ (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓)
6	3, 4, 5	3imtr3g 294	. 2 ⊢ (∀𝑥𝜑 → (¬ ∃𝑥𝜒 → ¬ ∃𝑥𝜓))
7	6	con4d 115	1 ⊢ (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1539  ∃wex 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811

This theorem depends on definitions:  df-bi 206  df-ex 1782

This theorem is referenced by:  alexbii  1835  exim  1836  eximdh  1867  19.29  1876  19.29r  1877  19.35  1880  19.25  1883  19.30  1884  19.40b  1891  exintr  1895  19.36imv  1948  speimfw  1967  aeveq  2059  sbequ2  2241  2ax6elem  2468  sb1  2476  dfeumo  2530  mo3  2557  mo4  2559  mopick  2620  2mo  2643  ssel  3955  ssopab2  5523  ssoprab2  7445  axextnd  10551  bj-2exim  35186  bj-exalimi  35207  bj-cbvalimt  35213  bj-eximcom  35217  bj-subst  35235  bj-gabss  35512  wl-mo3t  36138  pm10.56  42805  2exim  42814