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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 3672 | . 2 ⊢ ([𝐴 / 𝑦]∀𝑥𝜑 → 𝐴 ∈ V) | |
2 | sbcex 3672 | . . 3 ⊢ ([𝐴 / 𝑦]𝜑 → 𝐴 ∈ V) | |
3 | 2 | sps 2226 | . 2 ⊢ (∀𝑥[𝐴 / 𝑦]𝜑 → 𝐴 ∈ V) |
4 | dfsbcq2 3665 | . . 3 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ [𝐴 / 𝑦]∀𝑥𝜑)) | |
5 | dfsbcq2 3665 | . . . 4 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑦]𝜑 ↔ [𝐴 / 𝑦]𝜑)) | |
6 | 5 | albidv 2019 | . . 3 ⊢ (𝑧 = 𝐴 → (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑)) |
7 | sbal 2596 | . . 3 ⊢ ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑) | |
8 | 4, 6, 7 | vtoclbg 3483 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑)) |
9 | 1, 3, 8 | pm5.21nii 370 | 1 ⊢ ([𝐴 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∀wal 1654 = wceq 1656 [wsb 2067 ∈ wcel 2164 Vcvv 3414 [wsbc 3662 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-v 3416 df-sbc 3663 |
This theorem is referenced by: sbcabel 3741 sbcssg 4307 sbcfung 6151 bnj89 31332 bnj110 31470 bnj611 31530 bnj1000 31553 bj-sbeq 33416 bj-sbceqgALT 33417 sbcalf 34457 frege70 39066 frege77 39073 frege116 39112 frege118 39114 trsbc 39583 trsbcVD 39930 sbcssgVD 39936 |
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