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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 324 | . . . 4 ⊢ ((x = y ∧ φ) → (x = y ∧ ∀yφ)) |
3 | 2 | eximi 1488 | . . 3 ⊢ (∃x(x = y ∧ φ) → ∃x(x = y ∧ ∀yφ)) |
4 | equs5a 1672 | . . 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 1610 | . 2 ⊢ (∀x(x = y → φ) → ∃x(x = y ∧ φ)) | |
7 | 5, 6 | impbii 117 | 1 ⊢ (∃x(x = y ∧ φ) ↔ ∀x(x = y → φ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∀wal 1240 ∃wex 1378 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1333 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-11 1394 ax-4 1397 ax-i9 1420 ax-ial 1424 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: sb5f 1682 |
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