Theorem moimi 2603

Description: The at-most-one quantifier reverses implication. (Contributed by NM, 15-Feb-2006.) Remove use of ax-5 1911. (Revised by Steven Nguyen, 9-May-2023.)

Hypothesis

Ref	Expression
moimi.1	⊢ (𝜑 → 𝜓)

Assertion

Ref	Expression
moimi	⊢ (∃*𝑥𝜓 → ∃*𝑥𝜑)

Proof of Theorem moimi

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		moimi.1	. . . . 5 ⊢ (𝜑 → 𝜓)
2	1	imim1i 63	. . . 4 ⊢ ((𝜓 → 𝑥 = 𝑦) → (𝜑 → 𝑥 = 𝑦))
3	2	alimi 1813	. . 3 ⊢ (∀𝑥(𝜓 → 𝑥 = 𝑦) → ∀𝑥(𝜑 → 𝑥 = 𝑦))
4	3	eximi 1836	. 2 ⊢ (∃𝑦∀𝑥(𝜓 → 𝑥 = 𝑦) → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
5		df-mo 2598	. 2 ⊢ (∃*𝑥𝜓 ↔ ∃𝑦∀𝑥(𝜓 → 𝑥 = 𝑦))
6		df-mo 2598	. 2 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
7	4, 5, 6	3imtr4i 295	1 ⊢ (∃*𝑥𝜓 → ∃*𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1536  ∃wex 1781  ∃*wmo 2596

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 210  df-ex 1782  df-mo 2598

This theorem is referenced by:  mobii  2606  moa1  2610  moan  2611  moor  2613  mooran1  2614  mooran2  2615  moaneu  2685  2moexv  2689  2euexv  2693  2exeuv  2694  2moex  2702  2euex  2703  2exeu  2708  sndisj  5021  disjxsn  5023  axsepgfromrep  5165  fununmo  6371  funcnvsn  6374  nfunsn  6682  caovmo  7365  iunmapdisj  9434  brdom3  9939  brdom5  9940  brdom4  9941  nqerf  10341  shftfn  14424  2ndcdisj2  22062  plyexmo  24909  ajfuni  28642  funadj  29669  cnlnadjeui  29860  funressnvmo  43637