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Mirrors > Home > MPE Home > Th. List > sbcnestgf | Structured version Visualization version GIF version |
Description: Nest the composition of two substitutions. Usage of this theorem is discouraged because it depends on ax-13 2390. Use the weaker sbcnestgfw 4370 when possible. (Contributed by Mario Carneiro, 11-Nov-2016.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sbcnestgf | ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥𝜑) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsbcq 3774 | . . . . 5 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑)) | |
2 | csbeq1 3886 | . . . . . 6 ⊢ (𝑧 = 𝐴 → ⦋𝑧 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵) | |
3 | 2 | sbceq1d 3777 | . . . . 5 ⊢ (𝑧 = 𝐴 → ([⦋𝑧 / 𝑥⦌𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑)) |
4 | 1, 3 | bibi12d 348 | . . . 4 ⊢ (𝑧 = 𝐴 → (([𝑧 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝑧 / 𝑥⦌𝐵 / 𝑦]𝜑) ↔ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑))) |
5 | 4 | imbi2d 343 | . . 3 ⊢ (𝑧 = 𝐴 → ((∀𝑦Ⅎ𝑥𝜑 → ([𝑧 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝑧 / 𝑥⦌𝐵 / 𝑦]𝜑)) ↔ (∀𝑦Ⅎ𝑥𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑)))) |
6 | vex 3497 | . . . . 5 ⊢ 𝑧 ∈ V | |
7 | 6 | a1i 11 | . . . 4 ⊢ (∀𝑦Ⅎ𝑥𝜑 → 𝑧 ∈ V) |
8 | csbeq1a 3897 | . . . . . 6 ⊢ (𝑥 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵) | |
9 | 8 | sbceq1d 3777 | . . . . 5 ⊢ (𝑥 = 𝑧 → ([𝐵 / 𝑦]𝜑 ↔ [⦋𝑧 / 𝑥⦌𝐵 / 𝑦]𝜑)) |
10 | 9 | adantl 484 | . . . 4 ⊢ ((∀𝑦Ⅎ𝑥𝜑 ∧ 𝑥 = 𝑧) → ([𝐵 / 𝑦]𝜑 ↔ [⦋𝑧 / 𝑥⦌𝐵 / 𝑦]𝜑)) |
11 | nfnf1 2158 | . . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝜑 | |
12 | 11 | nfal 2342 | . . . 4 ⊢ Ⅎ𝑥∀𝑦Ⅎ𝑥𝜑 |
13 | nfa1 2155 | . . . . 5 ⊢ Ⅎ𝑦∀𝑦Ⅎ𝑥𝜑 | |
14 | nfcsb1v 3907 | . . . . . 6 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵 | |
15 | 14 | a1i 11 | . . . . 5 ⊢ (∀𝑦Ⅎ𝑥𝜑 → Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵) |
16 | sp 2182 | . . . . 5 ⊢ (∀𝑦Ⅎ𝑥𝜑 → Ⅎ𝑥𝜑) | |
17 | 13, 15, 16 | nfsbcd 3796 | . . . 4 ⊢ (∀𝑦Ⅎ𝑥𝜑 → Ⅎ𝑥[⦋𝑧 / 𝑥⦌𝐵 / 𝑦]𝜑) |
18 | 7, 10, 12, 17 | sbciedf 3813 | . . 3 ⊢ (∀𝑦Ⅎ𝑥𝜑 → ([𝑧 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝑧 / 𝑥⦌𝐵 / 𝑦]𝜑)) |
19 | 5, 18 | vtoclg 3567 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑦Ⅎ𝑥𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑))) |
20 | 19 | imp 409 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥𝜑) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1535 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 Ⅎwnfc 2961 Vcvv 3494 [wsbc 3772 ⦋csb 3883 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-13 2390 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-sbc 3773 df-csb 3884 |
This theorem is referenced by: csbnestgf 4376 sbcnestg 4377 |
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