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| Mirrors > Home > MPE Home > Th. List > Mathboxes > sbali | Structured version Visualization version GIF version | ||
| Description: Discard class substitution in a universal quantification when substituting the quantified variable, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.) |
| Ref | Expression |
|---|---|
| sbali.1 | ⊢ 𝐴 ∈ V |
| Ref | Expression |
|---|---|
| sbali | ⊢ ([𝐴 / 𝑥]∀𝑥𝜑 ↔ ∀𝑥𝜑) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sbali.1 | . 2 ⊢ 𝐴 ∈ V | |
| 2 | nfa1 2151 | . 2 ⊢ Ⅎ𝑥∀𝑥𝜑 | |
| 3 | 1, 2 | sbcgfi 3864 | 1 ⊢ ([𝐴 / 𝑥]∀𝑥𝜑 ↔ ∀𝑥𝜑) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∀wal 1538 ∈ 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-10 2141 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-sbc 3789 |
| This theorem is referenced by: (None) |
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