Theorem rspce 3624

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 2908	. . . 4 ⊢ Ⅎ𝑥𝐴
2		nfv 1913	. . . . 5 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵
3		rspc.1	. . . . 5 ⊢ Ⅎ𝑥𝜓
4	2, 3	nfan 1898	. . . 4 ⊢ Ⅎ𝑥(𝐴 ∈ 𝐵 ∧ 𝜓)
5		eleq1 2832	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
6		rspc.2	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
7	5, 6	anbi12d 631	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 ∧ 𝜑) ↔ (𝐴 ∈ 𝐵 ∧ 𝜓)))
8	1, 4, 7	spcegf 3605	. . 3 ⊢ (𝐴 ∈ 𝐵 → ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)))
9	8	anabsi5 668	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))
10		df-rex 3077	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))
11	9, 10	sylibr 234	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 𝜑)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537  ∃wex 1777  Ⅎwnf 1781   ∈ wcel 2108  ∃wrex 3076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-cleq 2732  df-clel 2819  df-nfc 2895  df-rex 3077

This theorem is referenced by:  reuop  6324  ac6c4  10550  infcvgaux1i  15905  iunmbl2  25611  gsumpart  33038  esumcvg  34050  ptrecube  37580  poimirlem24  37604  sdclem1  37703  uzwo4  44955  eliuniincex  45011  elrnmpt1sf  45096  iuneqfzuzlem  45249  uzublem  45345  uzub  45346  limsupubuzlem  45633  sge0gerp  46316  smflim  46698  reupr  47396