Theorem moimi 2540

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 1812	. . 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 2535	. 2 ⊢ (∃*𝑥𝜓 ↔ ∃𝑦∀𝑥(𝜓 → 𝑥 = 𝑦))
6		df-mo 2535	. 2 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
7	4, 5, 6	3imtr4i 292	1 ⊢ (∃*𝑥𝜓 → ∃*𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1539  ∃wex 1780  ∃*wmo 2533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810

This theorem depends on definitions:  df-bi 207  df-ex 1781  df-mo 2535

This theorem is referenced by:  mobii  2543  moa1  2546  moan  2547  moor  2549  mooran1  2550  mooran2  2551  moaneu  2618  2moexv  2622  2euexv  2626  2exeuv  2627  2moex  2635  2euex  2636  2exeu  2641  sndisj  5081  disjxsn  5083  axsepgfromrep  5230  fununmo  6528  funcnvsn  6531  nfunsn  6861  caovmo  7583  iunmapdisj  9914  brdom3  10419  brdom5  10420  brdom4  10421  nqerf  10821  shftfn  14980  2ndcdisj2  23372  plyexmo  26248  ajfuni  30839  funadj  31866  cnlnadjeui  32057  amosym1  36470  sinnpoly  46990  funressnvmo  47144