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Mirrors > Home > MPE Home > Th. List > equs45f | Structured version Visualization version GIF version |
Description: Two ways of expressing substitution when 𝑦 is not free in 𝜑. The implication "to the left" is equs4 2414 and does not require the nonfreeness hypothesis. Theorem sbalex 2234 replaces the nonfreeness hypothesis with a disjoint variable condition and equs5 2458 replaces it with a distinctor antecedent. (Contributed by NM, 25-Apr-2008.) (Revised by Mario Carneiro, 4-Oct-2016.) Usage of this theorem is discouraged because it depends on ax-13 2370. Use sbalex 2234 instead. (New usage is discouraged.) |
Ref | Expression |
---|---|
equs45f.1 | ⊢ Ⅎ𝑦𝜑 |
Ref | Expression |
---|---|
equs45f | ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equs45f.1 | . . . . . 6 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | nf5ri 2187 | . . . . 5 ⊢ (𝜑 → ∀𝑦𝜑) |
3 | 2 | anim2i 616 | . . . 4 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → (𝑥 = 𝑦 ∧ ∀𝑦𝜑)) |
4 | 3 | eximi 1836 | . . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑)) |
5 | equs5a 2455 | . . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
7 | equs4 2414 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | |
8 | 6, 7 | impbii 208 | 1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1538 ∃wex 1780 Ⅎwnf 1784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-10 2136 ax-12 2170 ax-13 2370 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-ex 1781 df-nf 1785 |
This theorem is referenced by: sb5f 2496 |
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