Theorem spcedv 2892

Description: Existential specialization, using implicit substitution, deduction version. (Contributed by RP, 12-Aug-2020.)

Hypotheses

Ref	Expression
spcedv.1	⊢ (𝜑 → 𝑋 ∈ V)
spcedv.2	⊢ (𝜑 → 𝜒)
spcedv.3	⊢ (𝑥 = 𝑋 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
spcedv	⊢ (𝜑 → ∃𝑥𝜓)

Distinct variable groups:   𝑥,𝑋   𝜒,𝑥

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

Proof of Theorem spcedv

Step	Hyp	Ref	Expression
1		spcedv.1	. 2 ⊢ (𝜑 → 𝑋 ∈ V)
2		spcedv.2	. 2 ⊢ (𝜑 → 𝜒)
3		spcedv.3	. . 3 ⊢ (𝑥 = 𝑋 → (𝜓 ↔ 𝜒))
4	3	spcegv 2891	. 2 ⊢ (𝑋 ∈ V → (𝜒 → ∃𝑥𝜓))
5	1, 2, 4	sylc 62	1 ⊢ (𝜑 → ∃𝑥𝜓)

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1395  ∃wex 1538   ∈ wcel 2200  Vcvv 2799

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801

This theorem is referenced by:  fprodseq  12089  gsumval2  13425