Theorem r19.9rmv 3333

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 2141	. . 3 ⊢ (𝑎 = 𝑦 → (𝑎 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
2	1	cbvexv 1836	. 2 ⊢ (∃𝑎 𝑎 ∈ 𝐴 ↔ ∃𝑦 𝑦 ∈ 𝐴)
3		eleq1 2141	. . . 4 ⊢ (𝑎 = 𝑥 → (𝑎 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
4	3	cbvexv 1836	. . 3 ⊢ (∃𝑎 𝑎 ∈ 𝐴 ↔ ∃𝑥 𝑥 ∈ 𝐴)
5		df-rex 2354	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
6		19.41v 1823	. . . . 5 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (∃𝑥 𝑥 ∈ 𝐴 ∧ 𝜑))
7	5, 6	bitri 182	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 𝑥 ∈ 𝐴 ∧ 𝜑))
8	7	baibr 862	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜑))
9	4, 8	sylbi 119	. 2 ⊢ (∃𝑎 𝑎 ∈ 𝐴 → (𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜑))
10	2, 9	sylbir 133	1 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜑))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103  ∃wex 1421   ∈ wcel 1433  ∃wrex 2349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-cleq 2074  df-clel 2077  df-rex 2354

This theorem is referenced by:  r19.45mv  3335  iunconstm  3686  fconstfvm  5400  ltexprlemloc  6797  lcmgcdlem  10459