Theorem spcimedv 3265

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		spcimdv.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
2		spcimedv.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜒 → 𝜓))
3	2	con3d 147	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (¬ 𝜓 → ¬ 𝜒))
4	1, 3	spcimdv 3263	. . 3 ⊢ (𝜑 → (∀𝑥 ¬ 𝜓 → ¬ 𝜒))
5	4	con2d 128	. 2 ⊢ (𝜑 → (𝜒 → ¬ ∀𝑥 ¬ 𝜓))
6		df-ex 1696	. 2 ⊢ (∃𝑥𝜓 ↔ ¬ ∀𝑥 ¬ 𝜓)
7	5, 6	syl6ibr 241	1 ⊢ (𝜑 → (𝜒 → ∃𝑥𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175

This theorem is referenced by:  hashf1rn  13004  hashf1rnOLD  13005  cshwsexa  13421  wwlktovfo  13549  uvcendim  20005  wlkiswwlk2  26225  wlknwwlknsur  26240  wlkiswwlksur  26247  wwlkextsur  26259  clwlkisclwwlklem2  26314  rtrclex  36943  clcnvlem  36949  iunrelexpuztr  37030  1wlkiswwlks2  41072  wlknwwlksnsur  41087  wlkwwlksur  41094  wwlksnextsur  41106  elwwlks2  41170  elwspths2spth  41171  clwlkclwwlklem1  41208