![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > sbcco3g | GIF version |
Description: Composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.) |
Ref | Expression |
---|---|
sbcco3g.1 | ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
sbcco3g | ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐶 / 𝑦]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcnestg 3019 | . 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑)) | |
2 | elex 2668 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
3 | nfcvd 2256 | . . . 4 ⊢ (𝐴 ∈ V → Ⅎ𝑥𝐶) | |
4 | sbcco3g.1 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
5 | 3, 4 | csbiegf 3009 | . . 3 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) |
6 | dfsbcq 2880 | . . 3 ⊢ (⦋𝐴 / 𝑥⦌𝐵 = 𝐶 → ([⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑 ↔ [𝐶 / 𝑦]𝜑)) | |
7 | 2, 5, 6 | 3syl 17 | . 2 ⊢ (𝐴 ∈ 𝑉 → ([⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑 ↔ [𝐶 / 𝑦]𝜑)) |
8 | 1, 7 | bitrd 187 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐶 / 𝑦]𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1314 ∈ wcel 1463 Vcvv 2657 [wsbc 2878 ⦋csb 2971 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-v 2659 df-sbc 2879 df-csb 2972 |
This theorem is referenced by: fzshftral 9781 |
Copyright terms: Public domain | W3C validator |