Theorem spcimedv 2825

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 1874	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜒 → 𝜓)))
4		spcimdv.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
5		nfv 1528	. . 3 ⊢ Ⅎ𝑥𝜒
6		nfcv 2319	. . 3 ⊢ Ⅎ𝑥𝐴
7	5, 6	spcimegft 2817	. 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 1351   = wceq 1353  ∃wex 1492   ∈ wcel 2148

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 2741

This theorem is referenced by:  rspcimedv  2845  fihashf1rn  10771