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Mirrors > Home > MPE Home > Th. List > hba1wOLD | Structured version Visualization version GIF version |
Description: Obsolete proof of hba1w 1974 as of 10-Oct-2021. (Contributed by NM, 9-Apr-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hbn1w.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
hba1wOLD | ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbn1w.1 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
2 | 1 | cbvalvw 1969 | . . . . . 6 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) |
3 | 2 | a1i 11 | . . . . 5 ⊢ (𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓)) |
4 | 3 | notbid 308 | . . . 4 ⊢ (𝑥 = 𝑦 → (¬ ∀𝑥𝜑 ↔ ¬ ∀𝑦𝜓)) |
5 | 4 | spw 1967 | . . 3 ⊢ (∀𝑥 ¬ ∀𝑥𝜑 → ¬ ∀𝑥𝜑) |
6 | 5 | con2i 134 | . 2 ⊢ (∀𝑥𝜑 → ¬ ∀𝑥 ¬ ∀𝑥𝜑) |
7 | 4 | hbn1w 1973 | . 2 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ ∀𝑥𝜑) |
8 | 1 | hbn1w 1973 | . . . 4 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑) |
9 | 8 | con1i 144 | . . 3 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥𝜑) |
10 | 9 | alimi 1739 | . 2 ⊢ (∀𝑥 ¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) |
11 | 6, 7, 10 | 3syl 18 | 1 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∀wal 1481 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 |
This theorem is referenced by: (None) |
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