Theorem r19.3rmv 3604

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 1577	. 2 ⊢ Ⅎ𝑥𝜑
2	1	r19.3rm 3602	1 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))

Syntax hints:   → wi 4   ↔ wb 105  ∃wex 1541   ∈ wcel 2205  ∀wral 2522

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-cleq 2227  df-clel 2230  df-ral 2527

This theorem is referenced by:  iinconstm  4005  exmidsssnc  4321  cnvpom  5310  ssfilem  7143  ssfilemd  7145  diffitest  7157  inffiexmid  7179  ctssexmid  7454  exmidonfinlem  7509  caucvgsrlemasr  8121  resqrexlemgt0  11730  rmodislmodlem  14624  rmodislmod  14625