Theorem mo2r 2130

Description: A condition which implies "at most one". (Contributed by Jim Kingdon, 2-Jul-2018.)

Hypothesis

Ref	Expression
mo2r.1	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
mo2r	⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃*𝑥𝜑)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem mo2r

Step	Hyp	Ref	Expression
1		mo2r.1	. . . . 5 ⊢ Ⅎ𝑦𝜑
2	1	nfri 1565	. . . 4 ⊢ (𝜑 → ∀𝑦𝜑)
3	2	eu3h 2123	. . 3 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
4	3	simplbi2com 1487	. 2 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → (∃𝑥𝜑 → ∃!𝑥𝜑))
5		df-mo 2081	. 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
6	4, 5	sylibr 134	1 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃*𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1393  Ⅎwnf 1506  ∃wex 1538  ∃!weu 2077  ∃*wmo 2078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081

This theorem is referenced by:  mo2icl  2982  rmo2ilem  3119  dffun5r  5330  frecuzrdgtcl  10642  frecuzrdgfunlem  10649