Theorem r19.9rmv 3586

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 2294	. . 3 ⊢ (𝑎 = 𝑦 → (𝑎 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
2	1	cbvexv 1967	. 2 ⊢ (∃𝑎 𝑎 ∈ 𝐴 ↔ ∃𝑦 𝑦 ∈ 𝐴)
3		eleq1 2294	. . . 4 ⊢ (𝑎 = 𝑥 → (𝑎 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
4	3	cbvexv 1967	. . 3 ⊢ (∃𝑎 𝑎 ∈ 𝐴 ↔ ∃𝑥 𝑥 ∈ 𝐴)
5		df-rex 2516	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
6		19.41v 1951	. . . . 5 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (∃𝑥 𝑥 ∈ 𝐴 ∧ 𝜑))
7	5, 6	bitri 184	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 𝑥 ∈ 𝐴 ∧ 𝜑))
8	7	baibr 927	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜑))
9	4, 8	sylbi 121	. 2 ⊢ (∃𝑎 𝑎 ∈ 𝐴 → (𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜑))
10	2, 9	sylbir 135	1 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜑))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∃wex 1540   ∈ wcel 2202  ∃wrex 2511

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-cleq 2224  df-clel 2227  df-rex 2516

This theorem is referenced by:  r19.45mv  3588  r19.44mv  3589  iunconstm  3978  fconstfvm  5871  frecabcl  6564  ltexprlemloc  7826  lcmgcdlem  12648  dvdsr02  14118