Theorem moim 2250

Description: "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 22-Apr-1995.)

Assertion

Ref	Expression
moim	⊢ (∀x(φ → ψ) → (∃*xψ → ∃*xφ))

Proof of Theorem moim

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		imim1 70	. . . 4 ⊢ ((φ → ψ) → ((ψ → x = y) → (φ → x = y)))
2	1	al2imi 1561	. . 3 ⊢ (∀x(φ → ψ) → (∀x(ψ → x = y) → ∀x(φ → x = y)))
3	2	eximdv 1622	. 2 ⊢ (∀x(φ → ψ) → (∃y∀x(ψ → x = y) → ∃y∀x(φ → x = y)))
4		nfv 1619	. . 3 ⊢ Ⅎyψ
5	4	mo2 2233	. 2 ⊢ (∃*xψ ↔ ∃y∀x(ψ → x = y))
6		nfv 1619	. . 3 ⊢ Ⅎyφ
7	6	mo2 2233	. 2 ⊢ (∃*xφ ↔ ∃y∀x(φ → x = y))
8	3, 5, 7	3imtr4g 261	1 ⊢ (∀x(φ → ψ) → (∃*xψ → ∃*xφ))

Syntax hints:   → wi 4  ∀wal 1540  ∃wex 1541   = wceq 1642  ∃*wmo 2205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209

This theorem is referenced by:  moimi  2251  euimmo  2253  moexex  2273  rmoim  3036  rmoimi2  3038  funmo  5126