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Mirrors > Home > MPE Home > Th. List > Mathboxes > ich2ex | Structured version Visualization version GIF version |
Description: Two setvar variables are always interchangeable when there are two existential quantifiers. (Contributed by SN, 23-Nov-2023.) |
Ref | Expression |
---|---|
ich2ex | ⊢ [𝑥⇄𝑦]∃𝑥∃𝑦𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfe1 2151 | . 2 ⊢ Ⅎ𝑥∃𝑥∃𝑦𝜑 | |
2 | excom 2166 | . . 3 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑) | |
3 | nfe1 2151 | . . 3 ⊢ Ⅎ𝑦∃𝑦∃𝑥𝜑 | |
4 | 2, 3 | nfxfr 1860 | . 2 ⊢ Ⅎ𝑦∃𝑥∃𝑦𝜑 |
5 | 1, 4 | ichf 44575 | 1 ⊢ [𝑥⇄𝑦]∃𝑥∃𝑦𝜑 |
Colors of variables: wff setvar class |
Syntax hints: ∃wex 1787 [wich 44570 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-10 2141 ax-11 2158 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-ex 1788 df-nf 1792 df-sb 2071 df-ich 44571 |
This theorem is referenced by: (None) |
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