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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 2154 | . 2 ⊢ Ⅎ𝑥∀𝑥𝜑 | |
3 | 1, 2 | sbcgfi 3851 | 1 ⊢ ([𝐴 / 𝑥]∀𝑥𝜑 ↔ ∀𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∀wal 1534 ∈ wcel 2113 Vcvv 3497 [wsbc 3775 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-sbc 3776 |
This theorem is referenced by: (None) |
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