Theorem elabd 3352

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 3294	. 2 ⊢ (𝑋 ∈ V → (𝜒 → ∃𝑥𝜓))
5	1, 2, 4	sylc 65	1 ⊢ (𝜑 → ∃𝑥𝜓)

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1483  ∃wex 1704   ∈ wcel 1990  Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202

This theorem is referenced by:  hasheqf1od  13144  setsexstruct2  15897  wwlksnextbij  26797  clrellem  37929  clcnvlem  37930  uspgrsprfo  41756  uspgrbispr  41759