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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 2290 and does not require the non-freeness hypothesis. Theorem sb56 2150 replaces the non-freeness hypothesis with a dv condition and equs5 2351 replaces it with a distinctor as antecedent. (Contributed by NM, 25-Apr-2008.) (Revised by Mario Carneiro, 4-Oct-2016.) |
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 2065 | . . . . 5 ⊢ (𝜑 → ∀𝑦𝜑) |
3 | 2 | anim2i 593 | . . . 4 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → (𝑥 = 𝑦 ∧ ∀𝑦𝜑)) |
4 | 3 | eximi 1762 | . . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑)) |
5 | equs5a 2348 | . . 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 2290 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | |
8 | 6, 7 | impbii 199 | 1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∀wal 1481 ∃wex 1704 Ⅎwnf 1708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-10 2019 ax-12 2047 ax-13 2246 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-ex 1705 df-nf 1710 |
This theorem is referenced by: sb5f 2386 |
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