Theorem spcimedv 3532

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 3530	. . 3 ⊢ (𝜑 → (∀𝑥 ¬ 𝜓 → ¬ 𝜒))
5	4	con2d 134	. 2 ⊢ (𝜑 → (𝜒 → ¬ ∀𝑥 ¬ 𝜓))
6		df-ex 1786	. 2 ⊢ (∃𝑥𝜓 ↔ ¬ ∀𝑥 ¬ 𝜓)
7	5, 6	syl6ibr 251	1 ⊢ (𝜑 → (𝜒 → ∃𝑥𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∀wal 1539   = wceq 1541  ∃wex 1785   ∈ wcel 2109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1544  df-ex 1786  df-sb 2071  df-clab 2717  df-clel 2817

This theorem is referenced by:  spc3egv  3540  hashf1rn  14048  cshwsexa  14518  wwlktovfo  14654  uvcendim  21035  wlkiswwlks2  28219  wwlksnextsurj  28244  elwwlks2  28310  elwspths2spth  28311  clwlkclwwlklem1  28342  sticksstones4  40085  rtrclex  41178  clcnvlem  41184  iunrelexpuztr  41280