Theorem r19.9rmv 3307

Description: Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 5-Aug-2018.)

Assertion

Ref	Expression
r19.9rmv	⊢ (∃y y ∈ A → (φ ↔ ∃x ∈ A φ))

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

Allowed substitution hint:   φ(y)

Proof of Theorem r19.9rmv

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		df-rex 2306	. . . . 5 ⊢ (∃x ∈ A φ ↔ ∃x(x ∈ A ∧ φ))
6		19.41v 1779	. . . . 5 ⊢ (∃x(x ∈ A ∧ φ) ↔ (∃x x ∈ A ∧ φ))
7	5, 6	bitri 173	. . . 4 ⊢ (∃x ∈ A φ ↔ (∃x x ∈ A ∧ φ))
8	7	baibr 828	. . 3 ⊢ (∃x x ∈ A → (φ ↔ ∃x ∈ A φ))
9	4, 8	sylbi 114	. 2 ⊢ (∃𝑎 𝑎 ∈ A → (φ ↔ ∃x ∈ A φ))
10	2, 9	sylbir 125	1 ⊢ (∃y y ∈ A → (φ ↔ ∃x ∈ A φ))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∃wex 1378   ∈ wcel 1390  ∃wrex 2301

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-ext 2019

This theorem depends on definitions:  df-bi 110  df-cleq 2030  df-clel 2033  df-rex 2306

This theorem is referenced by:  r19.45mv  3309  iunconstm  3656  fconstfvm  5322  ltexprlemloc  6581