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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 2119) but is used as an auxiliary axiom scheme to achieve metalogical completeness. Use its weak version alcomiw 2039 when it allows to avoid dependence on ax-11 2147. (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 1532 | . . 3 wff ∀𝑦𝜑 |
4 | vx | . . 3 setvar 𝑥 | |
5 | 3, 4 | wal 1532 | . 2 wff ∀𝑥∀𝑦𝜑 |
6 | 1, 4 | wal 1532 | . . 3 wff ∀𝑥𝜑 |
7 | 6, 2 | wal 1532 | . 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 2148 alcom 2149 hbal 2157 hbald 2158 hbsbwOLD 2162 nfald 2317 hbae 2426 hbaltg 35397 bj-hbalt 36152 bj-nnflemaa 36233 bj-nfald 36610 hbae-o 38369 axc711 38380 axc5c711 38384 ax12indalem 38411 ax12inda2ALT 38412 pm11.71 43828 axc5c4c711 43832 axc11next 43837 hbalg 43988 hbalgVD 44338 hbexgVD 44339 ichal 46800 |
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