Theorem cbvcsb 3538

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 2758	. . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐶
3		cbvcsb.2	. . . . 5 ⊢ Ⅎ𝑥𝐷
4	3	nfcri 2758	. . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐷
5		cbvcsb.3	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)
6	5	eleq2d 2687	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐶 ↔ 𝑧 ∈ 𝐷))
7	2, 4, 6	cbvsbc 3464	. . 3 ⊢ ([𝐴 / 𝑥]𝑧 ∈ 𝐶 ↔ [𝐴 / 𝑦]𝑧 ∈ 𝐷)
8	7	abbii 2739	. 2 ⊢ {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐶} = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐷}
9		df-csb 3534	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐶}
10		df-csb 3534	. 2 ⊢ ⦋𝐴 / 𝑦⦌𝐷 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐷}
11	8, 9, 10	3eqtr4i 2654	1 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑦⦌𝐷

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  {cab 2608  Ⅎwnfc 2751  [wsbc 3435  ⦋csb 3533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-sbc 3436  df-csb 3534

This theorem is referenced by:  cbvcsbv  3539  cbvsum  14425  cbvprod  14645  measiuns  30280  poimirlem26  33435  climinf2mpt  39946  climinfmpt  39947