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Mirrors > Home > MPE Home > Th. List > cbvex | Structured version Visualization version GIF version |
Description: Rule used to change bound variables, using implicit substitution. See cbvexv 2330, cbvexv1 2276, and cbvexvw 1993 for weaker versions. The latter two use fewer axioms. (Contributed by NM, 21-Jun-1993.) |
Ref | Expression |
---|---|
cbval.1 | ⊢ Ⅎ𝑦𝜑 |
cbval.2 | ⊢ Ⅎ𝑥𝜓 |
cbval.3 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvex | ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbval.1 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | nfn 1819 | . . . 4 ⊢ Ⅎ𝑦 ¬ 𝜑 |
3 | cbval.2 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
4 | 3 | nfn 1819 | . . . 4 ⊢ Ⅎ𝑥 ¬ 𝜓 |
5 | cbval.3 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
6 | 5 | notbid 310 | . . . 4 ⊢ (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓)) |
7 | 2, 4, 6 | cbval 2327 | . . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ∀𝑦 ¬ 𝜓) |
8 | alnex 1744 | . . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑) | |
9 | alnex 1744 | . . 3 ⊢ (∀𝑦 ¬ 𝜓 ↔ ¬ ∃𝑦𝜓) | |
10 | 7, 8, 9 | 3bitr3i 293 | . 2 ⊢ (¬ ∃𝑥𝜑 ↔ ¬ ∃𝑦𝜓) |
11 | 10 | con4bii 313 | 1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∀wal 1505 ∃wex 1742 Ⅎwnf 1746 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-11 2091 ax-12 2104 ax-13 2299 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-ex 1743 df-nf 1747 |
This theorem is referenced by: cbvexv 2330 exsbOLD 2420 sb8e 2482 mofOLD 2572 eufOLD 2589 cbveuALT 2632 cbvmoOLD 2633 |
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