Theorem rmoim 3036

Description: Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Assertion

Ref	Expression
rmoim	⊢ (∀x ∈ A (φ → ψ) → (∃*x ∈ A ψ → ∃*x ∈ A φ))

Proof of Theorem rmoim

Step	Hyp	Ref	Expression
1		df-ral 2620	. . 3 ⊢ (∀x ∈ A (φ → ψ) ↔ ∀x(x ∈ A → (φ → ψ)))
2		imdistan 670	. . . 4 ⊢ ((x ∈ A → (φ → ψ)) ↔ ((x ∈ A ∧ φ) → (x ∈ A ∧ ψ)))
3	2	albii 1566	. . 3 ⊢ (∀x(x ∈ A → (φ → ψ)) ↔ ∀x((x ∈ A ∧ φ) → (x ∈ A ∧ ψ)))
4	1, 3	bitri 240	. 2 ⊢ (∀x ∈ A (φ → ψ) ↔ ∀x((x ∈ A ∧ φ) → (x ∈ A ∧ ψ)))
5		moim 2250	. . 3 ⊢ (∀x((x ∈ A ∧ φ) → (x ∈ A ∧ ψ)) → (∃*x(x ∈ A ∧ ψ) → ∃*x(x ∈ A ∧ φ)))
6		df-rmo 2623	. . 3 ⊢ (∃*x ∈ A ψ ↔ ∃*x(x ∈ A ∧ ψ))
7		df-rmo 2623	. . 3 ⊢ (∃*x ∈ A φ ↔ ∃*x(x ∈ A ∧ φ))
8	5, 6, 7	3imtr4g 261	. 2 ⊢ (∀x((x ∈ A ∧ φ) → (x ∈ A ∧ ψ)) → (∃*x ∈ A ψ → ∃*x ∈ A φ))
9	4, 8	sylbi 187	1 ⊢ (∀x ∈ A (φ → ψ) → (∃*x ∈ A ψ → ∃*x ∈ A φ))

Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540   ∈ wcel 1710  ∃*wmo 2205  ∀wral 2615  ∃*wrmo 2618

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  df-ral 2620  df-rmo 2623

This theorem is referenced by:  rmoimia  3037  2rmorex  3041