Theorem r19.3rmv 3480

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 1505	. 2 ⊢ Ⅎ𝑥𝜑
2	1	r19.3rm 3478	1 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))

Syntax hints:   → wi 4   ↔ wb 104  ∃wex 1469   ∈ wcel 2125  ∀wral 2432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-cleq 2147  df-clel 2150  df-ral 2437

This theorem is referenced by:  iinconstm  3854  exmidsssnc  4159  cnvpom  5121  ssfilem  6809  diffitest  6821  inffiexmid  6840  ctssexmid  7072  exmidonfinlem  7107  caucvgsrlemasr  7689  resqrexlemgt0  10897