Theorem moimi 2544

Description: The at-most-one quantifier reverses implication. (Contributed by NM, 15-Feb-2006.) Remove use of ax-5 1910. (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 1811	. . 3 ⊢ (∀𝑥(𝜓 → 𝑥 = 𝑦) → ∀𝑥(𝜑 → 𝑥 = 𝑦))
4	3	eximi 1835	. 2 ⊢ (∃𝑦∀𝑥(𝜓 → 𝑥 = 𝑦) → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
5		df-mo 2539	. 2 ⊢ (∃*𝑥𝜓 ↔ ∃𝑦∀𝑥(𝜓 → 𝑥 = 𝑦))
6		df-mo 2539	. 2 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
7	4, 5, 6	3imtr4i 292	1 ⊢ (∃*𝑥𝜓 → ∃*𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1538  ∃wex 1779  ∃*wmo 2537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 207  df-ex 1780  df-mo 2539

This theorem is referenced by:  mobii  2547  moa1  2550  moan  2551  moor  2553  mooran1  2554  mooran2  2555  moaneu  2622  2moexv  2626  2euexv  2630  2exeuv  2631  2moex  2639  2euex  2640  2exeu  2645  sndisj  5111  disjxsn  5113  axsepgfromrep  5264  fununmo  6583  funcnvsn  6586  nfunsn  6918  caovmo  7644  iunmapdisj  10037  brdom3  10542  brdom5  10543  brdom4  10544  nqerf  10944  shftfn  15092  2ndcdisj2  23395  plyexmo  26273  ajfuni  30840  funadj  31867  cnlnadjeui  32058  amosym1  36444  funressnvmo  47074