Theorem sbcnestg 3185

Description: Nest the composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)

Assertion

Ref	Expression
sbcnestg	⊢ (A ∈ V → ([̣A / x]̣[̣B / y]̣φ ↔ [̣[A / x]B / y]̣φ))

Distinct variable group:   φ,x

Allowed substitution hints:   φ(y)   A(x,y)   B(x,y)   V(x,y)

Proof of Theorem sbcnestg

Step	Hyp	Ref	Expression
1		nfv 1619	. . 3 ⊢ Ⅎxφ
2	1	ax-gen 1546	. 2 ⊢ ∀yℲxφ
3		sbcnestgf 3183	. 2 ⊢ ((A ∈ V ∧ ∀yℲxφ) → ([̣A / x]̣[̣B / y]̣φ ↔ [̣[A / x]B / y]̣φ))
4	2, 3	mpan2 652	1 ⊢ (A ∈ V → ([̣A / x]̣[̣B / y]̣φ ↔ [̣[A / x]B / y]̣φ))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544   ∈ wcel 1710  [̣wsbc 3046  [csb 3136

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-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-sbc 3047  df-csb 3137

This theorem is referenced by:  sbcco3g  3191