Theorem elabd 2951

Description: Explicit demonstration the class {𝑥 ∣ 𝜓} is not empty by the example 𝑋. (Contributed by RP, 12-Aug-2020.)

Hypotheses

Ref	Expression
elab.xex	⊢ (𝜑 → 𝑋 ∈ V)
elab.xmaj	⊢ (𝜑 → 𝜒)
elab.xsub	⊢ (𝑥 = 𝑋 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
elabd	⊢ (𝜑 → ∃𝑥𝜓)

Distinct variable groups:   𝜒,𝑥   𝑥,𝑋

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

Proof of Theorem elabd

Step	Hyp	Ref	Expression
1		elab.xex	. 2 ⊢ (𝜑 → 𝑋 ∈ V)
2		elab.xmaj	. 2 ⊢ (𝜑 → 𝜒)
3		elab.xsub	. . 3 ⊢ (𝑥 = 𝑋 → (𝜓 ↔ 𝜒))
4	3	spcegv 2894	. 2 ⊢ (𝑋 ∈ V → (𝜒 → ∃𝑥𝜓))
5	1, 2, 4	sylc 62	1 ⊢ (𝜑 → ∃𝑥𝜓)

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1397  ∃wex 1540   ∈ wcel 2202  Vcvv 2802

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804

This theorem is referenced by:  uchoice  6300  elmpom  6403  en2prd  6992  dom1o  7002  ntrivcvgap0  12112  ssomct  13068  wlkvtxiedg  16199  wlkvtxiedgg  16200  dceqnconst  16685  dcapnconst  16686  gfsumval  16701