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Mirrors > Home > MPE Home > Th. List > hbalw | Structured version Visualization version GIF version |
Description: Weak version of hbal 2023. Uses only Tarski's FOL axiom schemes. Unlike hbal 2023, this theorem requires that 𝑥 and 𝑦 be distinct, i.e. not be bundled. (Contributed by NM, 19-Apr-2017.) |
Ref | Expression |
---|---|
hbalw.1 | ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓)) |
hbalw.2 | ⊢ (𝜑 → ∀𝑥𝜑) |
Ref | Expression |
---|---|
hbalw | ⊢ (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbalw.2 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | |
2 | 1 | alimi 1730 | . 2 ⊢ (∀𝑦𝜑 → ∀𝑦∀𝑥𝜑) |
3 | hbalw.1 | . . 3 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓)) | |
4 | 3 | alcomiw 1958 | . 2 ⊢ (∀𝑦∀𝑥𝜑 → ∀𝑥∀𝑦𝜑) |
5 | 2, 4 | syl 17 | 1 ⊢ (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 |
This theorem is referenced by: (None) |
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