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| Description: Change bound variables in a wff substitution. Usage of this theorem is discouraged because it depends on ax-13 2376. Use the weaker cbvsbcw 3820 when possible. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| cbvsbc.1 | ⊢ Ⅎ𝑦𝜑 | 
| cbvsbc.2 | ⊢ Ⅎ𝑥𝜓 | 
| cbvsbc.3 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | 
| Ref | Expression | 
|---|---|
| cbvsbc | ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | cbvsbc.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
| 2 | cbvsbc.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
| 3 | cbvsbc.3 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
| 4 | 1, 2, 3 | cbvab 2813 | . . 3 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} | 
| 5 | 4 | eleq2i 2832 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝐴 ∈ {𝑦 ∣ 𝜓}) | 
| 6 | df-sbc 3788 | . 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) | |
| 7 | df-sbc 3788 | . 2 ⊢ ([𝐴 / 𝑦]𝜓 ↔ 𝐴 ∈ {𝑦 ∣ 𝜓}) | |
| 8 | 5, 6, 7 | 3bitr4i 303 | 1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 Ⅎwnf 1782 ∈ wcel 2107 {cab 2713 [wsbc 3787 | 
| 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-sbc 3788 | 
| This theorem is referenced by: cbvsbcv 3823 cbvcsb 3909 | 
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