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Mirrors > Home > ILE Home > Th. List > equs45f | GIF version |
Description: Two ways of expressing substitution when 𝑦 is not free in 𝜑. (Contributed by NM, 25-Apr-2008.) |
Ref | Expression |
---|---|
equs45f.1 | ⊢ (𝜑 → ∀𝑦𝜑) |
Ref | Expression |
---|---|
equs45f | ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equs45f.1 | . . . . 5 ⊢ (𝜑 → ∀𝑦𝜑) | |
2 | 1 | anim2i 334 | . . . 4 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → (𝑥 = 𝑦 ∧ ∀𝑦𝜑)) |
3 | 2 | eximi 1531 | . . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑)) |
4 | equs5a 1715 | . . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)) | |
5 | 3, 4 | syl 14 | . 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
6 | equs4 1653 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | |
7 | 5, 6 | impbii 124 | 1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∀wal 1282 ∃wex 1421 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-11 1437 ax-4 1440 ax-i9 1463 ax-ial 1467 |
This theorem depends on definitions: df-bi 115 |
This theorem is referenced by: sb5f 1725 |
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