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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 36659.  Although both
     sides need a DV condition on 𝑥, 𝑡 (or as in bj-ax12v3 36687 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 36724.  Note that in the LHS, the reverse
     implication holds by equs4 2420 (or equs4v 1998 if a DV condition is added on
     𝑥,
𝑡 as in bj-ax12 36659), and the forward implication is sbalex 2241. The LHS can be read as saying that if there exists a variable equal to a given term witnessing a given formula, then all variables equal to that term also witness that formula. The equivalent form of the LHS using only primitive symbols is (∀𝑥(𝑥 = 𝑡 → 𝜑) ∨ ∀𝑥(𝑥 = 𝑡 → ¬ 𝜑)), which expresses that a given formula is true at all variables equal to a given term, or false at all these variables. An equivalent form of the LHS using only the existential quantifier 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 a formula and the other witnessing its negation. These equivalences do not hold in intuitionistic logic. The LHS should be the preferred form, and has the advantage of having no negation nor nested quantifiers. (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 36716 | . . . . 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 1838 | . . . 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 2143 | . . . . . 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 36625 | . . . . 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 2182 | . . . . 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 209 | . 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 1818 | . 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 206 ∧ wa 395 ∀wal 1537 ∃wex 1778 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-10 2140 ax-12 2176 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 | 
| This theorem is referenced by: (None) | 
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