Theorem spcimedv 2859

Description: Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
spcimdv.1	⊢ (𝜑 → 𝐴 ∈ 𝐵)
spcimedv.2	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜒 → 𝜓))

Assertion

Ref	Expression
spcimedv	⊢ (𝜑 → (𝜒 → ∃𝑥𝜓))

Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝜒,𝑥

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

Proof of Theorem spcimedv

Step	Hyp	Ref	Expression
1		spcimedv.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜒 → 𝜓))
2	1	ex 115	. . 3 ⊢ (𝜑 → (𝑥 = 𝐴 → (𝜒 → 𝜓)))
3	2	alrimiv 1897	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜒 → 𝜓)))
4		spcimdv.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
5		nfv 1551	. . 3 ⊢ Ⅎ𝑥𝜒
6		nfcv 2348	. . 3 ⊢ Ⅎ𝑥𝐴
7	5, 6	spcimegft 2851	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜒 → 𝜓)) → (𝐴 ∈ 𝐵 → (𝜒 → ∃𝑥𝜓)))
8	3, 4, 7	sylc 62	1 ⊢ (𝜑 → (𝜒 → ∃𝑥𝜓))

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1371   = wceq 1373  ∃wex 1515   ∈ wcel 2176

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774

This theorem is referenced by:  rspcimedv  2879  fihashf1rn  10933