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Mirrors > Home > MPE Home > Th. List > alcomiw | Structured version Visualization version GIF version |
Description: Weak version of alcom 2037. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.) |
Ref | Expression |
---|---|
alcomiw.1 | ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
alcomiw | ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alcomiw.1 | . . . . 5 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) | |
2 | 1 | biimpd 219 | . . . 4 ⊢ (𝑦 = 𝑧 → (𝜑 → 𝜓)) |
3 | 2 | cbvalivw 1934 | . . 3 ⊢ (∀𝑦𝜑 → ∀𝑧𝜓) |
4 | 3 | alimi 1739 | . 2 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑥∀𝑧𝜓) |
5 | ax-5 1839 | . 2 ⊢ (∀𝑥∀𝑧𝜓 → ∀𝑦∀𝑥∀𝑧𝜓) | |
6 | 1 | biimprd 238 | . . . . . 6 ⊢ (𝑦 = 𝑧 → (𝜓 → 𝜑)) |
7 | 6 | equcoms 1947 | . . . . 5 ⊢ (𝑧 = 𝑦 → (𝜓 → 𝜑)) |
8 | 7 | spimvw 1927 | . . . 4 ⊢ (∀𝑧𝜓 → 𝜑) |
9 | 8 | alimi 1739 | . . 3 ⊢ (∀𝑥∀𝑧𝜓 → ∀𝑥𝜑) |
10 | 9 | alimi 1739 | . 2 ⊢ (∀𝑦∀𝑥∀𝑧𝜓 → ∀𝑦∀𝑥𝜑) |
11 | 4, 5, 10 | 3syl 18 | 1 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → 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: hbalw 1977 ax11w 2007 bj-ssblem2 32631 |
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