Theorem spcedv 2827

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 2826	. 2 ⊢ (𝑋 ∈ V → (𝜒 → ∃𝑥𝜓))
5	1, 2, 4	sylc 62	1 ⊢ (𝜑 → ∃𝑥𝜓)

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1353  ∃wex 1492   ∈ wcel 2148  Vcvv 2738

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740

This theorem is referenced by:  fprodseq  11591