Theorem rspce 2902

Description: Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) (Revised by Mario Carneiro, 11-Oct-2016.)

Hypotheses

Ref	Expression
rspc.1	⊢ Ⅎ𝑥𝜓
rspc.2	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
rspce	⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 𝜑)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem rspce

Step	Hyp	Ref	Expression
1		nfcv 2372	. . . 4 ⊢ Ⅎ𝑥𝐴
2		nfv 1574	. . . . 5 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵
3		rspc.1	. . . . 5 ⊢ Ⅎ𝑥𝜓
4	2, 3	nfan 1611	. . . 4 ⊢ Ⅎ𝑥(𝐴 ∈ 𝐵 ∧ 𝜓)
5		eleq1 2292	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
6		rspc.2	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
7	5, 6	anbi12d 473	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 ∧ 𝜑) ↔ (𝐴 ∈ 𝐵 ∧ 𝜓)))
8	1, 4, 7	spcegf 2886	. . 3 ⊢ (𝐴 ∈ 𝐵 → ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)))
9	8	anabsi5 579	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))
10		df-rex 2514	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))
11	9, 10	sylibr 134	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 𝜑)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395  Ⅎwnf 1506  ∃wex 1538   ∈ wcel 2200  ∃wrex 2509

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-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801

This theorem is referenced by:  rspcev  2907  bezoutlemmain  12514