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Mirrors > Home > MPE Home > Th. List > cbvexvOLD | Structured version Visualization version GIF version |
Description: Obsolete version of cbvexv 2418 as of 11-Sep-2023. (Contributed by NM, 21-Jun-1993.) Remove dependency on ax-10 2144. (Revised by Wolf Lammen, 17-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cbvalv.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvexvOLD | ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvalv.1 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
2 | 1 | notbid 320 | . . . 4 ⊢ (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓)) |
3 | 2 | cbvalv 2417 | . . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ∀𝑦 ¬ 𝜓) |
4 | alnex 1781 | . . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑) | |
5 | alnex 1781 | . . 3 ⊢ (∀𝑦 ¬ 𝜓 ↔ ¬ ∃𝑦𝜓) | |
6 | 3, 4, 5 | 3bitr3i 303 | . 2 ⊢ (¬ ∃𝑥𝜑 ↔ ¬ ∃𝑦𝜓) |
7 | 6 | con4bii 323 | 1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∀wal 1534 ∃wex 1779 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-11 2160 ax-12 2176 ax-13 2389 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 df-nf 1784 |
This theorem is referenced by: (None) |
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