Theorem rmoan 3035

Description: Restricted "at most one" still holds when a conjunct is added. (Contributed by NM, 16-Jun-2017.)

Assertion

Ref	Expression
rmoan	⊢ (∃*x ∈ A φ → ∃*x ∈ A (ψ ∧ φ))

Proof of Theorem rmoan

Step	Hyp	Ref	Expression
1		moan 2255	. . 3 ⊢ (∃*x(x ∈ A ∧ φ) → ∃*x(ψ ∧ (x ∈ A ∧ φ)))
2		an12 772	. . . 4 ⊢ ((ψ ∧ (x ∈ A ∧ φ)) ↔ (x ∈ A ∧ (ψ ∧ φ)))
3	2	mobii 2240	. . 3 ⊢ (∃*x(ψ ∧ (x ∈ A ∧ φ)) ↔ ∃*x(x ∈ A ∧ (ψ ∧ φ)))
4	1, 3	sylib 188	. 2 ⊢ (∃*x(x ∈ A ∧ φ) → ∃*x(x ∈ A ∧ (ψ ∧ φ)))
5		df-rmo 2623	. 2 ⊢ (∃*x ∈ A φ ↔ ∃*x(x ∈ A ∧ φ))
6		df-rmo 2623	. 2 ⊢ (∃*x ∈ A (ψ ∧ φ) ↔ ∃*x(x ∈ A ∧ (ψ ∧ φ)))
7	4, 5, 6	3imtr4i 257	1 ⊢ (∃*x ∈ A φ → ∃*x ∈ A (ψ ∧ φ))

Syntax hints:   → wi 4   ∧ wa 358   ∈ 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)