Theorem cbvcsbw 3902

Description: Change bound variables in a class substitution. Interestingly, this does not require any bound variable conditions on 𝐴. Version of cbvcsb 3903 with a disjoint variable condition, which does not require ax-13 2369. (Contributed by Jeff Hankins, 13-Sep-2009.) Avoid ax-13 2369. (Revised by Gino Giotto, 10-Jan-2024.)

Hypotheses

Ref	Expression
cbvcsbw.1	⊢ Ⅎ𝑦𝐶
cbvcsbw.2	⊢ Ⅎ𝑥𝐷
cbvcsbw.3	⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)

Assertion

Ref	Expression
cbvcsbw	⊢ ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑦⦌𝐷

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem cbvcsbw

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvcsbw.1	. . . . 5 ⊢ Ⅎ𝑦𝐶
2	1	nfcri 2888	. . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐶
3		cbvcsbw.2	. . . . 5 ⊢ Ⅎ𝑥𝐷
4	3	nfcri 2888	. . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐷
5		cbvcsbw.3	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)
6	5	eleq2d 2817	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐶 ↔ 𝑧 ∈ 𝐷))
7	2, 4, 6	cbvsbcw 3810	. . 3 ⊢ ([𝐴 / 𝑥]𝑧 ∈ 𝐶 ↔ [𝐴 / 𝑦]𝑧 ∈ 𝐷)
8	7	abbii 2800	. 2 ⊢ {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐶} = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐷}
9		df-csb 3893	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐶}
10		df-csb 3893	. 2 ⊢ ⦋𝐴 / 𝑦⦌𝐷 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐷}
11	8, 9, 10	3eqtr4i 2768	1 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑦⦌𝐷

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2104  {cab 2707  Ⅎwnfc 2881  [wsbc 3776  ⦋csb 3892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-11 2152  ax-12 2169  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-sbc 3777  df-csb 3893

This theorem is referenced by:  cbvcsbv  3904  cbvsum  15645  cbvprod  15863  measiuns  33513  poimirlem26  36817  climinf2mpt  44728  climinfmpt  44729