Theorem csbnestg 4378

Description: Nest the composition of two substitutions. Usage of this theorem is discouraged because it depends on ax-13 2374. Use the weaker csbnestgw 4373 when possible. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.) (New usage is discouraged.)

Assertion

Ref	Expression
csbnestg	⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐶)

Distinct variable group:   𝑥,𝐶

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

Proof of Theorem csbnestg

Step	Hyp	Ref	Expression
1		nfcv 2895	. . 3 ⊢ Ⅎ𝑥𝐶
2	1	ax-gen 1796	. 2 ⊢ ∀𝑦Ⅎ𝑥𝐶
3		csbnestgf 4376	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥𝐶) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐶)
4	2, 3	mpan2 691	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐶)

Syntax hints:   → wi 4  ∀wal 1539   = wceq 1541   ∈ wcel 2113  Ⅎwnfc 2880  ⦋csb 3846

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-13 2374  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-v 3439  df-sbc 3738  df-csb 3847

This theorem is referenced by:  csbco3g  4380