Theorem sbcth2 3130

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 2680	. 2 ⊢ ∀x ∈ B φ
3		rspsbc 3125	. 2 ⊢ (A ∈ B → (∀x ∈ B φ → [̣A / x]̣φ))
4	2, 3	mpi 16	1 ⊢ (A ∈ B → [̣A / x]̣φ)

Syntax hints:   → wi 4   ∈ wcel 1710  ∀wral 2615  [̣wsbc 3047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-v 2862  df-sbc 3048

This theorem is referenced by: (None)