Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > cbvcsb | Structured version Visualization version GIF version |
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 2370. Use the weaker cbvcsbw 3853 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 2891 | . . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐶 |
3 | cbvcsb.2 | . . . . 5 ⊢ Ⅎ𝑥𝐷 | |
4 | 3 | nfcri 2891 | . . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐷 |
5 | cbvcsb.3 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷) | |
6 | 5 | eleq2d 2822 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐶 ↔ 𝑧 ∈ 𝐷)) |
7 | 2, 4, 6 | cbvsbc 3763 | . . 3 ⊢ ([𝐴 / 𝑥]𝑧 ∈ 𝐶 ↔ [𝐴 / 𝑦]𝑧 ∈ 𝐷) |
8 | 7 | abbii 2806 | . 2 ⊢ {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐶} = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐷} |
9 | df-csb 3844 | . 2 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐶} | |
10 | df-csb 3844 | . 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 1540 ∈ wcel 2105 {cab 2713 Ⅎwnfc 2884 [wsbc 3727 ⦋csb 3843 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-13 2370 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1543 df-ex 1781 df-nf 1785 df-sb 2067 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-sbc 3728 df-csb 3844 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |