Theorem rspcimedv 2958

Description: Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
rspcimdv.1	⊢ (φ → A ∈ B)
rspcimedv.2	⊢ ((φ ∧ x = A) → (χ → ψ))

Assertion

Ref	Expression
rspcimedv	⊢ (φ → (χ → ∃x ∈ B ψ))

Distinct variable groups:   x,A   x,B   φ,x   χ,x

Allowed substitution hint:   ψ(x)

Proof of Theorem rspcimedv

Step	Hyp	Ref	Expression
1		rspcimdv.1	. . . 4 ⊢ (φ → A ∈ B)
2		rspcimedv.2	. . . . 5 ⊢ ((φ ∧ x = A) → (χ → ψ))
3	2	con3d 125	. . . 4 ⊢ ((φ ∧ x = A) → (¬ ψ → ¬ χ))
4	1, 3	rspcimdv 2957	. . 3 ⊢ (φ → (∀x ∈ B ¬ ψ → ¬ χ))
5	4	con2d 107	. 2 ⊢ (φ → (χ → ¬ ∀x ∈ B ¬ ψ))
6		dfrex2 2628	. 2 ⊢ (∃x ∈ B ψ ↔ ¬ ∀x ∈ B ¬ ψ)
7	5, 6	syl6ibr 218	1 ⊢ (φ → (χ → ∃x ∈ B ψ))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2615  ∃wrex 2616

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 2479  df-ral 2620  df-rex 2621  df-v 2862

This theorem is referenced by:  rspcedv  2960