Theorem rspcdv 2959

Description: Restricted specialization, using implicit substitution. (Contributed by NM, 17-Feb-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)

Hypotheses

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

Assertion

Ref	Expression
rspcdv	⊢ (φ → (∀x ∈ B ψ → χ))

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

Allowed substitution hint:   ψ(x)

Proof of Theorem rspcdv

Step	Hyp	Ref	Expression
1		rspcdv.1	. 2 ⊢ (φ → A ∈ B)
2		rspcdv.2	. . 3 ⊢ ((φ ∧ x = A) → (ψ ↔ χ))
3	2	biimpd 198	. 2 ⊢ ((φ ∧ x = A) → (ψ → χ))
4	1, 3	rspcimdv 2957	1 ⊢ (φ → (∀x ∈ B ψ → χ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2615

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-v 2862

This theorem is referenced by: (None)