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| Mirrors > Home > MPE Home > Th. List > sbcal | Structured version Visualization version GIF version | ||
| Description: Move universal quantifier in and out of class substitution. (Contributed by NM, 31-Dec-2016.) (Revised by NM, 18-Aug-2018.) |
| Ref | Expression |
|---|---|
| sbcal | ⊢ ([𝐴 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sbcex 3798 | . 2 ⊢ ([𝐴 / 𝑦]∀𝑥𝜑 → 𝐴 ∈ V) | |
| 2 | sbcex 3798 | . . 3 ⊢ ([𝐴 / 𝑦]𝜑 → 𝐴 ∈ V) | |
| 3 | 2 | sps 2185 | . 2 ⊢ (∀𝑥[𝐴 / 𝑦]𝜑 → 𝐴 ∈ V) |
| 4 | dfsbcq2 3791 | . . 3 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ [𝐴 / 𝑦]∀𝑥𝜑)) | |
| 5 | dfsbcq2 3791 | . . . 4 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑦]𝜑 ↔ [𝐴 / 𝑦]𝜑)) | |
| 6 | 5 | albidv 1920 | . . 3 ⊢ (𝑧 = 𝐴 → (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑)) |
| 7 | sbal 2169 | . . 3 ⊢ ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑) | |
| 8 | 4, 6, 7 | vtoclbg 3557 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑)) |
| 9 | 1, 3, 8 | pm5.21nii 378 | 1 ⊢ ([𝐴 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∀wal 1538 = wceq 1540 [wsb 2064 ∈ wcel 2108 Vcvv 3480 [wsbc 3788 |
| 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-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-sbc 3789 |
| This theorem is referenced by: sbcabel 3878 sbcssg 4520 sbcfung 6590 bnj89 34735 bnj110 34872 bnj611 34932 bnj1000 34955 bj-sbeq 36902 bj-sbceqgALT 36903 sbcalf 38121 frege70 43946 frege77 43953 frege116 43992 frege118 43994 trsbc 44560 trsbcVD 44897 sbcssgVD 44903 |
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