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Mathbox for Giovanni Mascellani |
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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 2151 | . 2 ⊢ Ⅎ𝑥∃𝑥𝜑 | |
3 | 1, 2 | sbcgfi 3794 | 1 ⊢ ([𝐴 / 𝑥]∃𝑥𝜑 ↔ ∃𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∃wex 1781 ∈ 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-10 2142 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-sbc 3721 |
This theorem is referenced by: (None) |
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