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Mirrors > Home > MPE Home > Th. List > cbvalw | Structured version Visualization version GIF version |
Description: Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) |
Ref | Expression |
---|---|
cbvalw.1 | ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑) |
cbvalw.2 | ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓) |
cbvalw.3 | ⊢ (∀𝑦𝜓 → ∀𝑥∀𝑦𝜓) |
cbvalw.4 | ⊢ (¬ 𝜑 → ∀𝑦 ¬ 𝜑) |
cbvalw.5 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvalw | ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvalw.1 | . . 3 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑) | |
2 | cbvalw.2 | . . 3 ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓) | |
3 | cbvalw.5 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 3 | biimpd 232 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
5 | 1, 2, 4 | cbvaliw 2014 | . 2 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓) |
6 | cbvalw.3 | . . 3 ⊢ (∀𝑦𝜓 → ∀𝑥∀𝑦𝜓) | |
7 | cbvalw.4 | . . 3 ⊢ (¬ 𝜑 → ∀𝑦 ¬ 𝜑) | |
8 | 3 | biimprd 251 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜓 → 𝜑)) |
9 | 8 | equcoms 2028 | . . 3 ⊢ (𝑦 = 𝑥 → (𝜓 → 𝜑)) |
10 | 6, 7, 9 | cbvaliw 2014 | . 2 ⊢ (∀𝑦𝜓 → ∀𝑥𝜑) |
11 | 5, 10 | impbii 212 | 1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∀wal 1541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 |
This theorem is referenced by: cbvalvw 2044 hbn1fw 2053 |
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