Theorem rmo2i 3133

Description: Condition implying restricted "at most one." (Contributed by NM, 17-Jun-2017.)

Hypothesis

Ref	Expression
rmo2.1	⊢ Ⅎyφ

Assertion

Ref	Expression
rmo2i	⊢ (∃y ∈ A ∀x ∈ A (φ → x = y) → ∃*x ∈ A φ)

Distinct variable group:   x,y,A

Allowed substitution hints:   φ(x,y)

Proof of Theorem rmo2i

Step	Hyp	Ref	Expression
1		rexex 2674	. 2 ⊢ (∃y ∈ A ∀x ∈ A (φ → x = y) → ∃y∀x ∈ A (φ → x = y))
2		rmo2.1	. . 3 ⊢ Ⅎyφ
3	2	rmo2 3132	. 2 ⊢ (∃*x ∈ A φ ↔ ∃y∀x ∈ A (φ → x = y))
4	1, 3	sylibr 203	1 ⊢ (∃y ∈ A ∀x ∈ A (φ → x = y) → ∃*x ∈ A φ)

Syntax hints:   → wi 4  ∃wex 1541  Ⅎwnf 1544  ∀wral 2615  ∃wrex 2616  ∃*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-rex 2621  df-rmo 2623

This theorem is referenced by: (None)