Theorem r19.3rmv 3550

Description: Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 6-Aug-2018.)

Assertion

Ref	Expression
r19.3rmv	⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))

Distinct variable groups:   𝑥,𝐴   𝑦,𝐴   𝜑,𝑥

Allowed substitution hint:   𝜑(𝑦)

Proof of Theorem r19.3rmv

Step	Hyp	Ref	Expression
1		nfv 1550	. 2 ⊢ Ⅎ𝑥𝜑
2	1	r19.3rm 3548	1 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))

Syntax hints:   → wi 4   ↔ wb 105  ∃wex 1514   ∈ wcel 2175  ∀wral 2483

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-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-cleq 2197  df-clel 2200  df-ral 2488

This theorem is referenced by:  iinconstm  3935  exmidsssnc  4246  cnvpom  5224  ssfilem  6971  diffitest  6983  inffiexmid  7002  ctssexmid  7251  exmidonfinlem  7300  caucvgsrlemasr  7902  resqrexlemgt0  11273  rmodislmodlem  14054  rmodislmod  14055