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Mirrors > Home > MPE Home > Th. List > alcomw | Structured version Visualization version GIF version |
Description: Weak version of alcom 2160 and biconditional form of alcomimw 2042. Uses only Tarski's FOL axiom schemes. (Contributed by BTernaryTau, 28-Dec-2024.) |
Ref | Expression |
---|---|
alcomw.1 | ⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜓)) |
alcomw.2 | ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜒)) |
Ref | Expression |
---|---|
alcomw | ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑦∀𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alcomw.2 | . . 3 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜒)) | |
2 | 1 | alcomimw 2042 | . 2 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑) |
3 | alcomw.1 | . . 3 ⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜓)) | |
4 | 3 | alcomimw 2042 | . 2 ⊢ (∀𝑦∀𝑥𝜑 → ∀𝑥∀𝑦𝜑) |
5 | 2, 4 | impbii 209 | 1 ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑦∀𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∀wal 1535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1778 |
This theorem is referenced by: cgsex4gOLD 3539 unissb 4963 dftr2c 5286 cotrg 6139 cnvsym 6144 dffun2 6583 |
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