Theorem spcimedv 2938

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 125	. . . 4 ⊢ ((φ ∧ x = A) → (¬ ψ → ¬ χ))
4	1, 3	spcimdv 2936	. . 3 ⊢ (φ → (∀x ¬ ψ → ¬ χ))
5	4	con2d 107	. 2 ⊢ (φ → (χ → ¬ ∀x ¬ ψ))
6		df-ex 1542	. 2 ⊢ (∃xψ ↔ ¬ ∀x ¬ ψ)
7	5, 6	syl6ibr 218	1 ⊢ (φ → (χ → ∃xψ))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861

This theorem is referenced by: (None)