Theorem spcimedv 3553

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 153	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (¬ 𝜓 → ¬ 𝜒))
4	1, 3	spcimdv 3551	. . 3 ⊢ (𝜑 → (∀𝑥 ¬ 𝜓 → ¬ 𝜒))
5	4	con2d 135	. 2 ⊢ (𝜑 → (𝜒 → ¬ ∀𝑥 ¬ 𝜓))
6		df-ex 1809	. 2 ⊢ (∃𝑥𝜓 ↔ ¬ ∀𝑥 ¬ 𝜓)
7	5, 6	imbitrrdi 255	1 ⊢ (𝜑 → (𝜒 → ∃𝑥𝜓))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 400  ∀wal 1567   = wceq 1569  ∃wex 1808   ∈ wcel 2142

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-clel 2837

This theorem is used by:  spc3egv  3561  hashf1rn  14395  wwlktovfo  15002  uvcendim  22008  wlkiswwlks2  30235  wwlksnextsurj  30260  elwwlks2  30329  elwspths2spth  30330  clwlkclwwlklem1  30361  sticksstones4  42944  rtrclex  44371  clcnvlem  44377  iunrelexpuztr  44473