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Mirrors > Home > MPE Home > Th. List > excom13 | Structured version Visualization version GIF version |
Description: Swap 1st and 3rd existential quantifiers. (Contributed by NM, 9-Mar-1995.) |
Ref | Expression |
---|---|
excom13 | ⊢ (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑧∃𝑦∃𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | excom 2168 | . 2 ⊢ (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑦∃𝑥∃𝑧𝜑) | |
2 | excom 2168 | . . 3 ⊢ (∃𝑥∃𝑧𝜑 ↔ ∃𝑧∃𝑥𝜑) | |
3 | 2 | exbii 1855 | . 2 ⊢ (∃𝑦∃𝑥∃𝑧𝜑 ↔ ∃𝑦∃𝑧∃𝑥𝜑) |
4 | excom 2168 | . 2 ⊢ (∃𝑦∃𝑧∃𝑥𝜑 ↔ ∃𝑧∃𝑦∃𝑥𝜑) | |
5 | 1, 3, 4 | 3bitri 300 | 1 ⊢ (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑧∃𝑦∃𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∃wex 1787 |
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-11 2160 |
This theorem depends on definitions: df-bi 210 df-ex 1788 |
This theorem is referenced by: exrot3 2171 exrot4 2172 euotd 5412 elfuns 33987 fundcmpsurbijinj 44580 |
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