Theorem cbvcsb 3862

Description: Change bound variables in a class substitution. Interestingly, this does not require any bound variable conditions on 𝐴. Usage of this theorem is discouraged because it depends on ax-13 2404. Use the weaker cbvcsbw 3861 when possible. (Contributed by Jeff Hankins, 13-Sep-2009.) (Revised by Mario Carneiro, 11-Dec-2016.) (New usage is discouraged.)

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 2918	. . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐶
3		cbvcsb.2	. . . . 5 ⊢ Ⅎ𝑥𝐷
4	3	nfcri 2918	. . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐷
5		cbvcsb.3	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)
6	5	eleq2d 2850	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐶 ↔ 𝑧 ∈ 𝐷))
7	2, 4, 6	cbvsbc 3777	. . 3 ⊢ ([𝐴 / 𝑥]𝑧 ∈ 𝐶 ↔ [𝐴 / 𝑦]𝑧 ∈ 𝐷)
8	7	abbii 2830	. 2 ⊢ {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐶} = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐷}
9		df-csb 3852	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐶}
10		df-csb 3852	. 2 ⊢ ⦋𝐴 / 𝑦⦌𝐷 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐷}
11	8, 9, 10	3eqtr4i 2796	1 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑦⦌𝐷

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  {cab 2741  Ⅎwnfc 2911  [wsbc 3742  ⦋csb 3851

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-13 2404  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-sbc 3743  df-csb 3852

This theorem is used by: (None)