Theorem sbcth2 3883

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 3062	. 2 ⊢ ∀𝑥 ∈ 𝐵 𝜑
3		rspsbc 3878	. 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 2107  ∀wral 3060  [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-11 2156  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-ral 3061  df-sbc 3788

This theorem is referenced by: (None)