Theorem spcimedv 2767

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 114	. . 3 ⊢ (𝜑 → (𝑥 = 𝐴 → (𝜒 → 𝜓)))
3	2	alrimiv 1846	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜒 → 𝜓)))
4		spcimdv.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
5		nfv 1508	. . 3 ⊢ Ⅎ𝑥𝜒
6		nfcv 2279	. . 3 ⊢ Ⅎ𝑥𝐴
7	5, 6	spcimegft 2759	. 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 103  ∀wal 1329   = wceq 1331  ∃wex 1468   ∈ wcel 1480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683

This theorem is referenced by:  rspcimedv  2786  fihashf1rn  10528