Theorem cbvcsb 2856

Description: Change bound variables in a class substitution. Interestingly, this does not require any bound variable conditions on 𝐴. (Contributed by Jeff Hankins, 13-Sep-2009.) (Revised by Mario Carneiro, 11-Dec-2016.)

Hypotheses

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

Assertion

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

Proof of Theorem cbvcsb

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvcsb.1	. . . . 5 ⊢ Ⅎ𝑦𝐶
2	1	nfcri 2172	. . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐶
3		cbvcsb.2	. . . . 5 ⊢ Ⅎ𝑥𝐷
4	3	nfcri 2172	. . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐷
5		cbvcsb.3	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)
6	5	eleq2d 2107	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐶 ↔ 𝑧 ∈ 𝐷))
7	2, 4, 6	cbvsbc 2791	. . 3 ⊢ ([𝐴 / 𝑥]𝑧 ∈ 𝐶 ↔ [𝐴 / 𝑦]𝑧 ∈ 𝐷)
8	7	abbii 2153	. 2 ⊢ {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐶} = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐷}
9		df-csb 2853	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐶}
10		df-csb 2853	. 2 ⊢ ⦋𝐴 / 𝑦⦌𝐷 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐷}
11	8, 9, 10	3eqtr4i 2070	1 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑦⦌𝐷

Syntax hints:   → wi 4   = wceq 1243   ∈ wcel 1393  {cab 2026  Ⅎwnfc 2165  [wsbc 2764  ⦋csb 2852

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-sbc 2765  df-csb 2853

This theorem is referenced by:  cbvcsbv  2857