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Mirrors > Home > ILE Home > Th. List > excom13 | 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 1554 | . 2 ⊢ (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑦∃𝑥∃𝑧𝜑) | |
2 | excom 1554 | . . 3 ⊢ (∃𝑥∃𝑧𝜑 ↔ ∃𝑧∃𝑥𝜑) | |
3 | 2 | exbii 1496 | . 2 ⊢ (∃𝑦∃𝑥∃𝑧𝜑 ↔ ∃𝑦∃𝑧∃𝑥𝜑) |
4 | excom 1554 | . 2 ⊢ (∃𝑦∃𝑧∃𝑥𝜑 ↔ ∃𝑧∃𝑦∃𝑥𝜑) | |
5 | 1, 3, 4 | 3bitri 195 | 1 ⊢ (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑧∃𝑦∃𝑥𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 ∃wex 1381 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-4 1400 ax-ial 1427 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: exrot3 1580 exrot4 1581 euotd 3991 |
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