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| Description: Weak version of ax-11 2156 from which we can prove any ax-11 2156 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. Unlike ax-11 2156, this theorem requires that 𝑥 and 𝑦 be distinct i.e. are not bundled. It is an alias of alcomimw 2041 introduced for labeling consistency. (Contributed by NM, 10-Apr-2017.) Use alcomimw 2041 instead. (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| ax11w.1 | ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) | 
| Ref | Expression | 
|---|---|
| ax11w | ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ax11w.1 | . 2 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) | |
| 2 | 1 | alcomimw 2041 | 1 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1537 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 | 
| This theorem is referenced by: (None) | 
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