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Mirrors > Home > MPE Home > Th. List > ax11w | Structured version Visualization version GIF version |
Description: Weak version of ax-11 2157 from which we can prove any ax-11 2157 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. Unlike ax-11 2157, this theorem requires that 𝑥 and 𝑦 be distinct i.e. are not bundled. It is an alias of alcomiw 2049 introduced for labeling consistency. (Contributed by NM, 10-Apr-2017.) Use alcomiw 2049 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 | alcomiw 2049 | 1 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1786 |
This theorem is referenced by: (None) |
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