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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-substax12 | Structured version Visualization version GIF version |
Description: Equivalent form of the
axiom of substitution bj-ax12 36186. Although both
sides need a DV condition on 𝑥, 𝑡 (or as in bj-ax12v3 36215 on
𝑡,
𝜑) to hold, their
equivalence holds without DV conditions. The
forward implication is proved in modal (K4) while the reverse implication
is proved in modal (T5). The LHS has the advantage of not involving
nested quantifiers on the same variable. Its metaweakening is proved from
the core axiom schemes in bj-substw 36252. Note that in the LHS, the reverse
implication holds by equs4 2409 (or equs4v 1995 if a DV condition is added on
𝑥,
𝑡 as in bj-ax12 36186), and the forward implication is sbalex 2230.
The LHS can be read as saying that if there exists a setvar equal to a given term witnessing 𝜑, then all setvars equal to that term also witness 𝜑. An equivalent suggestive form for the LHS is ¬ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) ∧ ∃𝑥(𝑥 = 𝑡 ∧ ¬ 𝜑)), which expresses that there can be no two variables both equal to a given term, one witnessing 𝜑 and the other witnessing ¬ 𝜑. (Contributed by BJ, 21-May-2024.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-substax12 | ⊢ ((∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)) ↔ ∀𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-modal4 36244 | . . . . 5 ⊢ (∀𝑥(𝑥 = 𝑡 → 𝜑) → ∀𝑥∀𝑥(𝑥 = 𝑡 → 𝜑)) | |
2 | 1 | imim2i 16 | . . . 4 ⊢ ((∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)) → (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥∀𝑥(𝑥 = 𝑡 → 𝜑))) |
3 | 19.38 1833 | . . . 4 ⊢ ((∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥∀𝑥(𝑥 = 𝑡 → 𝜑)) → ∀𝑥((𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑))) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ ((∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)) → ∀𝑥((𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑))) |
5 | hbe1a 2132 | . . . . . 6 ⊢ (∃𝑥∀𝑥(𝑥 = 𝑡 → 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)) | |
6 | 5, 1 | syl 17 | . . . . 5 ⊢ (∃𝑥∀𝑥(𝑥 = 𝑡 → 𝜑) → ∀𝑥∀𝑥(𝑥 = 𝑡 → 𝜑)) |
7 | bj-exlimg 36152 | . . . . 5 ⊢ ((∃𝑥∀𝑥(𝑥 = 𝑡 → 𝜑) → ∀𝑥∀𝑥(𝑥 = 𝑡 → 𝜑)) → (∀𝑥((𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)) → (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥∀𝑥(𝑥 = 𝑡 → 𝜑)))) | |
8 | 6, 7 | ax-mp 5 | . . . 4 ⊢ (∀𝑥((𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)) → (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥∀𝑥(𝑥 = 𝑡 → 𝜑))) |
9 | sp 2171 | . . . . 5 ⊢ (∀𝑥∀𝑥(𝑥 = 𝑡 → 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)) | |
10 | 9 | imim2i 16 | . . . 4 ⊢ ((∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥∀𝑥(𝑥 = 𝑡 → 𝜑)) → (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑))) |
11 | 8, 10 | syl 17 | . . 3 ⊢ (∀𝑥((𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)) → (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑))) |
12 | 4, 11 | impbii 208 | . 2 ⊢ ((∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)) ↔ ∀𝑥((𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑))) |
13 | impexp 449 | . . 3 ⊢ (((𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)) ↔ (𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))) | |
14 | 13 | albii 1813 | . 2 ⊢ (∀𝑥((𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)) ↔ ∀𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))) |
15 | 12, 14 | bitri 274 | 1 ⊢ ((∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)) ↔ ∀𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∀wal 1531 ∃wex 1773 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-10 2129 ax-12 2166 |
This theorem depends on definitions: df-bi 206 df-an 395 df-ex 1774 |
This theorem is referenced by: (None) |
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