Theorem sbcthdv 3062

Description: Deduction version of sbcth 3061. (Contributed by NM, 30-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Hypothesis

Ref	Expression
sbcthdv.1	⊢ (φ → ψ)

Assertion

Ref	Expression
sbcthdv	⊢ ((φ ∧ A ∈ V) → [̣A / x]̣ψ)

Distinct variable group:   φ,x

Allowed substitution hints:   ψ(x)   A(x)   V(x)

Proof of Theorem sbcthdv

Step	Hyp	Ref	Expression
1		sbcthdv.1	. . 3 ⊢ (φ → ψ)
2	1	alrimiv 1631	. 2 ⊢ (φ → ∀xψ)
3		spsbc 3059	. 2 ⊢ (A ∈ V → (∀xψ → [̣A / x]̣ψ))
4	2, 3	mpan9 455	1 ⊢ ((φ ∧ A ∈ V) → [̣A / x]̣ψ)

Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540   ∈ wcel 1710  [̣wsbc 3047

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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2862  df-sbc 3048

This theorem is referenced by: (None)