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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 3730 | . 2 ⊢ ([𝐴 / 𝑦]∀𝑥𝜑 → 𝐴 ∈ V) | |
2 | sbcex 3730 | . . 3 ⊢ ([𝐴 / 𝑦]𝜑 → 𝐴 ∈ V) | |
3 | 2 | sps 2182 | . 2 ⊢ (∀𝑥[𝐴 / 𝑦]𝜑 → 𝐴 ∈ V) |
4 | dfsbcq2 3723 | . . 3 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ [𝐴 / 𝑦]∀𝑥𝜑)) | |
5 | dfsbcq2 3723 | . . . 4 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑦]𝜑 ↔ [𝐴 / 𝑦]𝜑)) | |
6 | 5 | albidv 1921 | . . 3 ⊢ (𝑧 = 𝐴 → (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑)) |
7 | sbal 2163 | . . 3 ⊢ ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑) | |
8 | 4, 6, 7 | vtoclbg 3517 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑)) |
9 | 1, 3, 8 | pm5.21nii 383 | 1 ⊢ ([𝐴 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∀wal 1536 = wceq 1538 [wsb 2069 ∈ wcel 2111 Vcvv 3441 [wsbc 3720 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-sbc 3721 |
This theorem is referenced by: sbcabel 3807 sbcssg 4421 sbcfung 6348 bnj89 32101 bnj110 32240 bnj611 32300 bnj1000 32323 bj-sbeq 34342 bj-sbceqgALT 34343 sbcalf 35552 frege70 40634 frege77 40641 frege116 40680 frege118 40682 trsbc 41246 trsbcVD 41583 sbcssgVD 41589 |
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