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Mirrors > Home > MPE Home > Th. List > ax-11 | Structured version Visualization version GIF version |
Description: Axiom of Quantifier Commutation. This axiom says universal quantifiers can be swapped. Axiom scheme C6' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Lemma 12 of [Monk2] p. 109 and Axiom C5-3 of [Monk2] p. 113. This axiom scheme is logically redundant (see ax11w 2007) but is used as an auxiliary axiom scheme to achieve metalogical completeness. (Contributed by NM, 12-Mar-1993.) |
Ref | Expression |
---|---|
ax-11 | ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wph | . . . 4 wff 𝜑 | |
2 | vy | . . . 4 setvar 𝑦 | |
3 | 1, 2 | wal 1481 | . . 3 wff ∀𝑦𝜑 |
4 | vx | . . 3 setvar 𝑥 | |
5 | 3, 4 | wal 1481 | . 2 wff ∀𝑥∀𝑦𝜑 |
6 | 1, 4 | wal 1481 | . . 3 wff ∀𝑥𝜑 |
7 | 6, 2 | wal 1481 | . 2 wff ∀𝑦∀𝑥𝜑 |
8 | 5, 7 | wi 4 | 1 wff (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑) |
Colors of variables: wff setvar class |
This axiom is referenced by: alcoms 2035 hbal 2036 alcom 2037 hbald 2041 nfald 2165 hbae 2315 hbaltg 31713 bj-hbalt 32671 hbae-o 34188 axc711 34199 axc5c711 34203 ax12indalem 34230 ax12inda2ALT 34231 pm11.71 38597 axc5c4c711 38602 axc11next 38607 hbalg 38771 hbalgVD 39141 hbexgVD 39142 |
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