Theorem elabd 3321

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 3267	. 2 ⊢ (𝑋 ∈ V → (𝜒 → ∃𝑥𝜓))
5	1, 2, 4	sylc 63	1 ⊢ (𝜑 → ∃𝑥𝜓)

Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175

This theorem is referenced by:  hasheqf1od  13006  sizeusglecusg  26014  clrellem  36948  clcnvlem  36949  wwlksnextbij  41108