Theorem r19.3rm 3304

Description: Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 19-Dec-2018.)

Hypothesis

Ref	Expression
r19.3rm.1	⊢ Ⅎxφ

Assertion

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

Distinct variable groups:   x,A   y,A

Allowed substitution hints:   φ(x,y)

Proof of Theorem r19.3rm

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2097	. . 3 ⊢ (𝑎 = y → (𝑎 ∈ A ↔ y ∈ A))
2	1	cbvexv 1792	. 2 ⊢ (∃𝑎 𝑎 ∈ A ↔ ∃y y ∈ A)
3		eleq1 2097	. . . 4 ⊢ (𝑎 = x → (𝑎 ∈ A ↔ x ∈ A))
4	3	cbvexv 1792	. . 3 ⊢ (∃𝑎 𝑎 ∈ A ↔ ∃x x ∈ A)
5		biimt 230	. . . 4 ⊢ (∃x x ∈ A → (φ ↔ (∃x x ∈ A → φ)))
6		df-ral 2305	. . . . 5 ⊢ (∀x ∈ A φ ↔ ∀x(x ∈ A → φ))
7		r19.3rm.1	. . . . . 6 ⊢ Ⅎxφ
8	7	19.23 1565	. . . . 5 ⊢ (∀x(x ∈ A → φ) ↔ (∃x x ∈ A → φ))
9	6, 8	bitri 173	. . . 4 ⊢ (∀x ∈ A φ ↔ (∃x x ∈ A → φ))
10	5, 9	syl6bbr 187	. . 3 ⊢ (∃x x ∈ A → (φ ↔ ∀x ∈ A φ))
11	4, 10	sylbi 114	. 2 ⊢ (∃𝑎 𝑎 ∈ A → (φ ↔ ∀x ∈ A φ))
12	2, 11	sylbir 125	1 ⊢ (∃y y ∈ A → (φ ↔ ∀x ∈ A φ))

Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1240  Ⅎwnf 1346  ∃wex 1378   ∈ wcel 1390  ∀wral 2300

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-cleq 2030  df-clel 2033  df-ral 2305

This theorem is referenced by:  r19.28m  3305  r19.3rmv  3306  r19.27m  3310  indstr  8312