Theorem rmoimi2 3038

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

Hypothesis

Ref	Expression
rmoimi2.1	⊢ ∀x((x ∈ A ∧ φ) → (x ∈ B ∧ ψ))

Assertion

Ref	Expression
rmoimi2	⊢ (∃*x ∈ B ψ → ∃*x ∈ A φ)

Proof of Theorem rmoimi2

Step	Hyp	Ref	Expression
1		rmoimi2.1	. . 3 ⊢ ∀x((x ∈ A ∧ φ) → (x ∈ B ∧ ψ))
2		moim 2250	. . 3 ⊢ (∀x((x ∈ A ∧ φ) → (x ∈ B ∧ ψ)) → (∃*x(x ∈ B ∧ ψ) → ∃*x(x ∈ A ∧ φ)))
3	1, 2	ax-mp 5	. 2 ⊢ (∃*x(x ∈ B ∧ ψ) → ∃*x(x ∈ A ∧ φ))
4		df-rmo 2623	. 2 ⊢ (∃*x ∈ B ψ ↔ ∃*x(x ∈ B ∧ ψ))
5		df-rmo 2623	. 2 ⊢ (∃*x ∈ A φ ↔ ∃*x(x ∈ A ∧ φ))
6	3, 4, 5	3imtr4i 257	1 ⊢ (∃*x ∈ B ψ → ∃*x ∈ A φ)

Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540   ∈ wcel 1710  ∃*wmo 2205  ∃*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-rmo 2623

This theorem is referenced by: (None)