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Mirrors > Home > MPE Home > Th. List > cbvexdw | Structured version Visualization version GIF version |
Description: Deduction used to change bound variables, using implicit substitution. Version of cbvexd 2401 with a disjoint variable condition, which does not require ax-13 2365. (Contributed by NM, 2-Jan-2002.) Avoid ax-13 2365. (Revised by Gino Giotto, 10-Jan-2024.) |
Ref | Expression |
---|---|
cbvaldw.1 | ⊢ Ⅎ𝑦𝜑 |
cbvaldw.2 | ⊢ (𝜑 → Ⅎ𝑦𝜓) |
cbvaldw.3 | ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) |
Ref | Expression |
---|---|
cbvexdw | ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvaldw.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
2 | cbvaldw.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑦𝜓) | |
3 | 2 | nfnd 1853 | . . . 4 ⊢ (𝜑 → Ⅎ𝑦 ¬ 𝜓) |
4 | cbvaldw.3 | . . . . 5 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) | |
5 | notbi 319 | . . . . 5 ⊢ ((𝜓 ↔ 𝜒) ↔ (¬ 𝜓 ↔ ¬ 𝜒)) | |
6 | 4, 5 | imbitrdi 250 | . . . 4 ⊢ (𝜑 → (𝑥 = 𝑦 → (¬ 𝜓 ↔ ¬ 𝜒))) |
7 | 1, 3, 6 | cbvaldw 2328 | . . 3 ⊢ (𝜑 → (∀𝑥 ¬ 𝜓 ↔ ∀𝑦 ¬ 𝜒)) |
8 | alnex 1775 | . . 3 ⊢ (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓) | |
9 | alnex 1775 | . . 3 ⊢ (∀𝑦 ¬ 𝜒 ↔ ¬ ∃𝑦𝜒) | |
10 | 7, 8, 9 | 3bitr3g 313 | . 2 ⊢ (𝜑 → (¬ ∃𝑥𝜓 ↔ ¬ ∃𝑦𝜒)) |
11 | 10 | con4bid 317 | 1 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∀wal 1531 ∃wex 1773 Ⅎwnf 1777 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-11 2146 ax-12 2163 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-ex 1774 df-nf 1778 |
This theorem is referenced by: bj-gabima 36326 bj-opabco 36575 wl-mo2t 36949 |
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