Theorem sbcthdv 3803

Description: Deduction version of sbcth 3802. (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 1926	. 2 ⊢ (𝜑 → ∀𝑥𝜓)
3		spsbc 3800	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜓 → [𝐴 / 𝑥]𝜓))
4	2, 3	mpan9 506	1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → [𝐴 / 𝑥]𝜓)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1537   ∈ wcel 2107  [wsbc 3787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-sbc 3788

This theorem is referenced by: (None)