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Mirrors > Home > MPE Home > Th. List > hbe1w | Structured version Visualization version GIF version |
Description: Weak version of hbe1 2144. See comments for ax10w 2130. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.) |
Ref | Expression |
---|---|
hbn1w.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
hbe1w | ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ex 1782 | . 2 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑) | |
2 | hbn1w.1 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
3 | 2 | notbid 321 | . . 3 ⊢ (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓)) |
4 | 3 | hbn1w 2053 | . 2 ⊢ (¬ ∀𝑥 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑) |
5 | 1, 4 | hbxfrbi 1826 | 1 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∀wal 1536 ∃wex 1781 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 |
This theorem is referenced by: (None) |
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