Theorem sbcth2 3846

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 3087	. 2 ⊢ ∀𝑥 ∈ 𝐵 𝜑
3		rspsbc 3841	. 2 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → [𝐴 / 𝑥]𝜑))
4	2, 3	mpi 21	1 ⊢ (𝐴 ∈ 𝐵 → [𝐴 / 𝑥]𝜑)

Syntax hints:   → wi 4   ∈ wcel 2149  ∀wral 3085  [wsbc 3753

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-11 2198  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-sbc 3754

This theorem is referenced by: (None)