Theorem sbcnestg 4386

Description: Nest the composition of two substitutions. Usage of this theorem is discouraged because it depends on ax-13 2401. Use the weaker sbcnestgw 4381 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 1947	. . 3 ⊢ Ⅎ𝑥𝜑
2	1	ax-gen 1828	. 2 ⊢ ∀𝑦Ⅎ𝑥𝜑
3		sbcnestgf 4384	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥𝜑) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑))
4	2, 3	mpan2 704	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568  Ⅎwnf 1816   ∈ wcel 2145  [wsbc 3739  ⦋csb 3847

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-13 2401  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-v 3452  df-sbc 3740  df-csb 3848

This theorem is used by:  sbcco3g  4388