Theorem aleximi 1865

Description: A variant of al2imi 1848: instead of applying ∀𝑥 quantifiers to the final implication, replace them with ∃𝑥. A shorter proof is possible using nfa1 2189, sps 2224 and eximd 2255, 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 153	. . . 4 ⊢ (𝜑 → (¬ 𝜒 → ¬ 𝜓))
3	2	al2imi 1848	. . 3 ⊢ (∀𝑥𝜑 → (∀𝑥 ¬ 𝜒 → ∀𝑥 ¬ 𝜓))
4		alnex 1814	. . 3 ⊢ (∀𝑥 ¬ 𝜒 ↔ ¬ ∃𝑥𝜒)
5		alnex 1814	. . 3 ⊢ (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓)
6	3, 4, 5	3imtr3g 298	. 2 ⊢ (∀𝑥𝜑 → (¬ ∃𝑥𝜒 → ¬ ∃𝑥𝜓))
7	6	con4d 116	1 ⊢ (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-ex 1813

This theorem is used by:  alexbii  1866  exim  1867  eximdh  1897  19.29  1906  19.29r  1907  19.35  1910  19.25  1913  19.30  1914  19.40b  1921  exintr  1925  19.36imv  1978  speimfw  1996  aeveq  2091  sbequ2  2287  2ax6elem  2504  sb1  2512  dfeumo  2566  mo3  2594  mo4  2596  mopick  2655  2mo  2678  ssel  3932  ssrexv  4008  axprlem4  5399  ssopab2  5533  ssoprab2  7484  elirrv  9562  axextnd  10587  axnulregtco  37024  bj-2exim  37256  bj-exalimi  37271  bj-eximcom  37272  bj-subst  37316  bj-gabss  37604  wl-mo3t  38264  wl-eujustlem1  38276  pm10.56  45113  2exim  45122