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Mirrors > Home > MPE Home > Th. List > Mathboxes > ax-c4 | Structured version Visualization version GIF version |
Description: Axiom of Quantified
Implication. This axiom moves a universal quantifier
from outside to inside an implication, quantifying 𝜓. Notice that
𝑥 must not be a free variable in the
antecedent of the quantified
implication, and we express this by binding 𝜑 to "protect" the
axiom
from a 𝜑 containing a free 𝑥. Axiom
scheme C4' in [Megill]
p. 448 (p. 16 of the preprint). It is a special case of Lemma 5 of
[Monk2] p. 108 and Axiom 5 of [Mendelson] p. 69.
This axiom is obsolete and should no longer be used. It is proved above as Theorem axc4 2315. (Contributed by NM, 3-Jan-1993.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ax-c4 | ⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wph | . . . . 5 wff 𝜑 | |
2 | vx | . . . . 5 setvar 𝑥 | |
3 | 1, 2 | wal 1537 | . . . 4 wff ∀𝑥𝜑 |
4 | wps | . . . 4 wff 𝜓 | |
5 | 3, 4 | wi 4 | . . 3 wff (∀𝑥𝜑 → 𝜓) |
6 | 5, 2 | wal 1537 | . 2 wff ∀𝑥(∀𝑥𝜑 → 𝜓) |
7 | 4, 2 | wal 1537 | . . 3 wff ∀𝑥𝜓 |
8 | 3, 7 | wi 4 | . 2 wff (∀𝑥𝜑 → ∀𝑥𝜓) |
9 | 6, 8 | wi 4 | 1 wff (∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓)) |
Colors of variables: wff setvar class |
This axiom is referenced by: ax4fromc4 36916 ax10fromc7 36917 equid1 36921 |
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