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Mirrors > Home > MPE Home > Th. List > Mathboxes > sbexi | Structured version Visualization version GIF version |
Description: Discard class substitution in an existential quantification when substituting the quantified variable, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.) |
Ref | Expression |
---|---|
sbexi.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
sbexi | ⊢ ([𝐴 / 𝑥]∃𝑥𝜑 ↔ ∃𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbexi.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | nfe1 2145 | . 2 ⊢ Ⅎ𝑥∃𝑥𝜑 | |
3 | 1, 2 | sbcgfi 3845 | 1 ⊢ ([𝐴 / 𝑥]∃𝑥𝜑 ↔ ∃𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∃wex 1771 ∈ wcel 2105 Vcvv 3492 [wsbc 3769 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-sbc 3770 |
This theorem is referenced by: (None) |
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