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Mirrors > Home > NFE Home > Th. List > cbvex | GIF version |
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
cbvex.1 | ⊢ Ⅎyφ |
cbvex.2 | ⊢ Ⅎxψ |
cbvex.3 | ⊢ (x = y → (φ ↔ ψ)) |
Ref | Expression |
---|---|
cbvex | ⊢ (∃xφ ↔ ∃yψ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvex.1 | . . . . 5 ⊢ Ⅎyφ | |
2 | 1 | nfn 1793 | . . . 4 ⊢ Ⅎy ¬ φ |
3 | cbvex.2 | . . . . 5 ⊢ Ⅎxψ | |
4 | 3 | nfn 1793 | . . . 4 ⊢ Ⅎx ¬ ψ |
5 | cbvex.3 | . . . . 5 ⊢ (x = y → (φ ↔ ψ)) | |
6 | 5 | notbid 285 | . . . 4 ⊢ (x = y → (¬ φ ↔ ¬ ψ)) |
7 | 2, 4, 6 | cbval 1984 | . . 3 ⊢ (∀x ¬ φ ↔ ∀y ¬ ψ) |
8 | 7 | notbii 287 | . 2 ⊢ (¬ ∀x ¬ φ ↔ ¬ ∀y ¬ ψ) |
9 | df-ex 1542 | . 2 ⊢ (∃xφ ↔ ¬ ∀x ¬ φ) | |
10 | df-ex 1542 | . 2 ⊢ (∃yψ ↔ ¬ ∀y ¬ ψ) | |
11 | 8, 9, 10 | 3bitr4i 268 | 1 ⊢ (∃xφ ↔ ∃yψ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 176 ∀wal 1540 ∃wex 1541 Ⅎwnf 1544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 |
This theorem depends on definitions: df-bi 177 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 |
This theorem is referenced by: cbvexv 2003 cbvex2 2005 exsb 2130 euf 2210 mo 2226 cbvmo 2241 mopick 2266 clelab 2473 issetf 2864 eqvincf 2967 rexab2 3003 euabsn 3792 eluniab 3903 cbvopab1 4632 cbvopab2 4633 cbvopab1s 4634 opeliunxp 4820 dfdmf 4905 dfrnf 4962 cbvoprab1 5567 cbvoprab2 5568 |
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