Theorem sbcthdv 3418

Description: Deduction version of sbcth 3417. (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 1842	. 2 ⊢ (𝜑 → ∀𝑥𝜓)
3		spsbc 3415	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜓 → [𝐴 / 𝑥]𝜓))
4	2, 3	mpan9 485	1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → [𝐴 / 𝑥]𝜓)

Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473   ∈ wcel 1977  [wsbc 3402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-an 385  df-tru 1478  df-ex 1696  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175  df-sbc 3403

This theorem is referenced by: (None)