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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 36056. Although both
sides need a DV condition on 𝑥, 𝑡 (or as in bj-ax12v3 36085 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 36122. Note that in the LHS, the reverse
implication holds by equs4 2410 (or equs4v 1996 if a DV condition is added on
𝑥,
𝑡 as in bj-ax12 36056), and the forward implication is sbalex 2228.
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 36114 | . . . . 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 1834 | . . . 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 2133 | . . . . . 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 36022 | . . . . 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 2169 | . . . . 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 450 | . . 3 ⊢ (((𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)) ↔ (𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))) | |
14 | 13 | albii 1814 | . 2 ⊢ (∀𝑥((𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)) ↔ ∀𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))) |
15 | 12, 14 | bitri 275 | 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 395 ∀wal 1532 ∃wex 1774 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-10 2130 ax-12 2164 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1775 |
This theorem is referenced by: (None) |
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