Theorem elabd 2919

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 2862	. 2 ⊢ (𝑋 ∈ V → (𝜒 → ∃𝑥𝜓))
5	1, 2, 4	sylc 62	1 ⊢ (𝜑 → ∃𝑥𝜓)

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1373  ∃wex 1516   ∈ wcel 2177  Vcvv 2773

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775

This theorem is referenced by:  uchoice  6230  en2prd  6916  ntrivcvgap0  11904  ssomct  12860  dceqnconst  16073  dcapnconst  16074