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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 4388 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by NM, 27-Nov-2005.) Avoid ax-13 2371. (Revised by Gino Giotto, 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 4381 | . 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑)) | |
2 | elex 3462 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
3 | nfcvd 2905 | . . . 4 ⊢ (𝐴 ∈ V → Ⅎ𝑥𝐶) | |
4 | sbcco3gw.1 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
5 | 3, 4 | csbiegf 3890 | . . 3 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) |
6 | dfsbcq 3742 | . . 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 205 = wceq 1542 ∈ wcel 2107 Vcvv 3444 [wsbc 3740 ⦋csb 3856 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-v 3446 df-sbc 3741 df-csb 3857 |
This theorem is referenced by: fzshftral 13535 2rexfrabdioph 41162 3rexfrabdioph 41163 4rexfrabdioph 41164 6rexfrabdioph 41165 7rexfrabdioph 41166 |
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