Theorem rspcsbela 3196

Description: Special case related to rspsbc 3125. (Contributed by NM, 10-Dec-2005.) (Proof shortened by Eric Schmidt, 17-Jan-2007.)

Assertion

Ref	Expression
rspcsbela	⊢ ((A ∈ B ∧ ∀x ∈ B C ∈ D) → [A / x]C ∈ D)

Distinct variable groups:   x,B   x,D

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

Proof of Theorem rspcsbela

Step	Hyp	Ref	Expression
1		rspsbc 3125	. . 3 ⊢ (A ∈ B → (∀x ∈ B C ∈ D → [̣A / x]̣C ∈ D))
2		sbcel1g 3156	. . 3 ⊢ (A ∈ B → ([̣A / x]̣C ∈ D ↔ [A / x]C ∈ D))
3	1, 2	sylibd 205	. 2 ⊢ (A ∈ B → (∀x ∈ B C ∈ D → [A / x]C ∈ D))
4	3	imp 418	1 ⊢ ((A ∈ B ∧ ∀x ∈ B C ∈ D) → [A / x]C ∈ D)

This proof depends on syntax axioms:   → wi 4   ∧ wa 358   ∈ wcel 1710  ∀wral 2615  [̣wsbc 3047  [csb 3137

This proof depends on 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 proof 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  df-csb 3138

This theorem is used by: (None)