Theorem sbcth2 3893

Description: A substitution into a theorem. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)

Hypothesis

Ref	Expression
sbcth2.1	⊢ (𝑥 ∈ 𝐵 → 𝜑)

Assertion

Ref	Expression
sbcth2	⊢ (𝐴 ∈ 𝐵 → [𝐴 / 𝑥]𝜑)

Distinct variable group:   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem sbcth2

Step	Hyp	Ref	Expression
1		sbcth2.1	. . 3 ⊢ (𝑥 ∈ 𝐵 → 𝜑)
2	1	rgen 3061	. 2 ⊢ ∀𝑥 ∈ 𝐵 𝜑
3		rspsbc 3888	. 2 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → [𝐴 / 𝑥]𝜑))
4	2, 3	mpi 20	1 ⊢ (𝐴 ∈ 𝐵 → [𝐴 / 𝑥]𝜑)

Syntax hints:   → wi 4   ∈ wcel 2106  ∀wral 3059  [wsbc 3791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-11 2155  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-sbc 3792

This theorem is referenced by: (None)