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| Mirrors > Home > MPE Home > Th. List > sbcco3gw | Structured version Visualization version GIF version | ||
| Description: Composition of two substitutions. Version of sbcco3g 4430 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by NM, 27-Nov-2005.) Avoid ax-13 2377. (Revised by GG, 26-Jan-2024.) | 
| Ref | Expression | 
|---|---|
| sbcco3gw.1 | ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | 
| Ref | Expression | 
|---|---|
| sbcco3gw | ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐶 / 𝑦]𝜑)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sbcnestgw 4423 | . 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑)) | |
| 2 | elex 3501 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
| 3 | nfcvd 2906 | . . . 4 ⊢ (𝐴 ∈ V → Ⅎ𝑥𝐶) | |
| 4 | sbcco3gw.1 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
| 5 | 3, 4 | csbiegf 3932 | . . 3 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) | 
| 6 | dfsbcq 3790 | . . 3 ⊢ (⦋𝐴 / 𝑥⦌𝐵 = 𝐶 → ([⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑 ↔ [𝐶 / 𝑦]𝜑)) | |
| 7 | 2, 5, 6 | 3syl 18 | . 2 ⊢ (𝐴 ∈ 𝑉 → ([⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑 ↔ [𝐶 / 𝑦]𝜑)) | 
| 8 | 1, 7 | bitrd 279 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐶 / 𝑦]𝜑)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 Vcvv 3480 [wsbc 3788 ⦋csb 3899 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-v 3482 df-sbc 3789 df-csb 3900 | 
| This theorem is referenced by: fzshftral 13655 2rexfrabdioph 42807 3rexfrabdioph 42808 4rexfrabdioph 42809 6rexfrabdioph 42810 7rexfrabdioph 42811 | 
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