Theorem spcimedv 3524

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 152	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (¬ 𝜓 → ¬ 𝜒))
4	1, 3	spcimdv 3522	. . 3 ⊢ (𝜑 → (∀𝑥 ¬ 𝜓 → ¬ 𝜒))
5	4	con2d 134	. 2 ⊢ (𝜑 → (𝜒 → ¬ ∀𝑥 ¬ 𝜓))
6		df-ex 1784	. 2 ⊢ (∃𝑥𝜓 ↔ ¬ ∀𝑥 ¬ 𝜓)
7	5, 6	syl6ibr 251	1 ⊢ (𝜑 → (𝜒 → ∃𝑥𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∀wal 1537   = wceq 1539  ∃wex 1783   ∈ wcel 2108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-clel 2817

This theorem is referenced by:  spc3egv  3532  hashf1rn  13995  cshwsexa  14465  wwlktovfo  14601  uvcendim  20964  wlkiswwlks2  28141  wwlksnextsurj  28166  elwwlks2  28232  elwspths2spth  28233  clwlkclwwlklem1  28264  sticksstones4  40033  rtrclex  41114  clcnvlem  41120  iunrelexpuztr  41216