Theorem sbcnestg 4425

Description: Nest the composition of two substitutions. Usage of this theorem is discouraged because it depends on ax-13 2370. Use the weaker sbcnestgw 4420 when possible. (Contributed by NM, 27-Nov-2005.) (Proof shortened by Mario Carneiro, 11-Nov-2016.) (New usage is discouraged.)

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 1916	. . 3 ⊢ Ⅎ𝑥𝜑
2	1	ax-gen 1796	. 2 ⊢ ∀𝑦Ⅎ𝑥𝜑
3		sbcnestgf 4423	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥𝜑) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑))
4	2, 3	mpan2 688	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1538  Ⅎwnf 1784   ∈ wcel 2105  [wsbc 3777  ⦋csb 3893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-13 2370  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-v 3475  df-sbc 3778  df-csb 3894

This theorem is referenced by:  sbcco3g  4427