Theorem csbcom 4348

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 3800	. . . 4 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝑧 ∈ 𝐶 ↔ [𝐵 / 𝑦][𝐴 / 𝑥]𝑧 ∈ 𝐶)
2		sbcel2 4346	. . . . 5 ⊢ ([𝐵 / 𝑦]𝑧 ∈ 𝐶 ↔ 𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐶)
3	2	sbcbii 3772	. . . 4 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝑧 ∈ 𝐶 ↔ [𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐶)
4		sbcel2 4346	. . . . 5 ⊢ ([𝐴 / 𝑥]𝑧 ∈ 𝐶 ↔ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝐶)
5	4	sbcbii 3772	. . . 4 ⊢ ([𝐵 / 𝑦][𝐴 / 𝑥]𝑧 ∈ 𝐶 ↔ [𝐵 / 𝑦]𝑧 ∈ ⦋𝐴 / 𝑥⦌𝐶)
6	1, 3, 5	3bitr3i 300	. . 3 ⊢ ([𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐶 ↔ [𝐵 / 𝑦]𝑧 ∈ ⦋𝐴 / 𝑥⦌𝐶)
7		sbcel2 4346	. . 3 ⊢ ([𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐶 ↔ 𝑧 ∈ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶)
8		sbcel2 4346	. . 3 ⊢ ([𝐵 / 𝑦]𝑧 ∈ ⦋𝐴 / 𝑥⦌𝐶 ↔ 𝑧 ∈ ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝐶)
9	6, 7, 8	3bitr3i 300	. 2 ⊢ (𝑧 ∈ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 ↔ 𝑧 ∈ ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝐶)
10	9	eqriv 2735	1 ⊢ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝐶

Syntax hints:   = wceq 1539   ∈ wcel 2108  [wsbc 3711  ⦋csb 3828

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-nul 4254

This theorem is referenced by:  ovmpos  7399  fvmpocurryd  8058  f1od2  30958