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| 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 2376. Use the weaker cbvcsbw 3908 when possible. (Contributed by Jeff Hankins, 13-Sep-2009.) (Revised by Mario Carneiro, 11-Dec-2016.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| cbvcsb.1 | ⊢ Ⅎ𝑦𝐶 | 
| cbvcsb.2 | ⊢ Ⅎ𝑥𝐷 | 
| cbvcsb.3 | ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷) | 
| Ref | Expression | 
|---|---|
| cbvcsb | ⊢ ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑦⦌𝐷 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | cbvcsb.1 | . . . . 5 ⊢ Ⅎ𝑦𝐶 | |
| 2 | 1 | nfcri 2896 | . . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐶 | 
| 3 | cbvcsb.2 | . . . . 5 ⊢ Ⅎ𝑥𝐷 | |
| 4 | 3 | nfcri 2896 | . . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐷 | 
| 5 | cbvcsb.3 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷) | |
| 6 | 5 | eleq2d 2826 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐶 ↔ 𝑧 ∈ 𝐷)) | 
| 7 | 2, 4, 6 | cbvsbc 3822 | . . 3 ⊢ ([𝐴 / 𝑥]𝑧 ∈ 𝐶 ↔ [𝐴 / 𝑦]𝑧 ∈ 𝐷) | 
| 8 | 7 | abbii 2808 | . 2 ⊢ {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐶} = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐷} | 
| 9 | df-csb 3899 | . 2 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐶} | |
| 10 | df-csb 3899 | . 2 ⊢ ⦋𝐴 / 𝑦⦌𝐷 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐷} | |
| 11 | 8, 9, 10 | 3eqtr4i 2774 | 1 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑦⦌𝐷 | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 {cab 2713 Ⅎwnfc 2889 [wsbc 3787 ⦋csb 3898 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-13 2376 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-sbc 3788 df-csb 3899 | 
| This theorem is referenced by: (None) | 
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