Theorem csbcom 4369

Description: Commutative law for double substitution into a class. (Contributed by NM, 14-Nov-2005.) (Revised by NM, 18-Aug-2018.)

Assertion

Ref	Expression
csbcom	⊢ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝐶

Distinct variable groups:   𝑦,𝐴   𝑥,𝐵   𝑥,𝑦

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

Proof of Theorem csbcom

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbccom 3818	. . . 4 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝑧 ∈ 𝐶 ↔ [𝐵 / 𝑦][𝐴 / 𝑥]𝑧 ∈ 𝐶)
2		sbcel2 4367	. . . . 5 ⊢ ([𝐵 / 𝑦]𝑧 ∈ 𝐶 ↔ 𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐶)
3	2	sbcbii 3794	. . . 4 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝑧 ∈ 𝐶 ↔ [𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐶)
4		sbcel2 4367	. . . . 5 ⊢ ([𝐴 / 𝑥]𝑧 ∈ 𝐶 ↔ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝐶)
5	4	sbcbii 3794	. . . 4 ⊢ ([𝐵 / 𝑦][𝐴 / 𝑥]𝑧 ∈ 𝐶 ↔ [𝐵 / 𝑦]𝑧 ∈ ⦋𝐴 / 𝑥⦌𝐶)
6	1, 3, 5	3bitr3i 301	. . 3 ⊢ ([𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐶 ↔ [𝐵 / 𝑦]𝑧 ∈ ⦋𝐴 / 𝑥⦌𝐶)
7		sbcel2 4367	. . 3 ⊢ ([𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐶 ↔ 𝑧 ∈ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶)
8		sbcel2 4367	. . 3 ⊢ ([𝐵 / 𝑦]𝑧 ∈ ⦋𝐴 / 𝑥⦌𝐶 ↔ 𝑧 ∈ ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝐶)
9	6, 7, 8	3bitr3i 301	. 2 ⊢ (𝑧 ∈ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 ↔ 𝑧 ∈ ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝐶)
10	9	eqriv 2730	1 ⊢ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝐶

Syntax hints:   = wceq 1541   ∈ wcel 2113  [wsbc 3737  ⦋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-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  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  df-dif 3901  df-nul 4283

This theorem is referenced by:  ovmpos  7500  fvmpocurryd  8207  f1od2  32706