Theorem sbcthdv 3020

Description: Deduction version of sbcth 3019. (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 1898	. 2 ⊢ (𝜑 → ∀𝑥𝜓)
3		spsbc 3017	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜓 → [𝐴 / 𝑥]𝜓))
4	2, 3	mpan9 281	1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → [𝐴 / 𝑥]𝜓)

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1371   ∈ wcel 2178  [wsbc 3005

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-v 2778  df-sbc 3006

This theorem is referenced by: (None)