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Mirrors > Home > MPE Home > Th. List > ax12w | Structured version Visualization version GIF version |
Description: Weak version of ax-12 2171 from which we can prove any ax-12 2171 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. An instance of the first hypothesis will normally require that 𝑥 and 𝑦 be distinct (unless 𝑥 does not occur in 𝜑). For an example of how the hypotheses can be eliminated when we substitute an expression without wff variables for 𝜑, see ax12wdemo 2131. (Contributed by NM, 10-Apr-2017.) |
Ref | Expression |
---|---|
ax12w.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
ax12w.2 | ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜒)) |
Ref | Expression |
---|---|
ax12w | ⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax12w.2 | . . 3 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜒)) | |
2 | 1 | spw 2037 | . 2 ⊢ (∀𝑦𝜑 → 𝜑) |
3 | ax12w.1 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 3 | ax12wlem 2128 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
5 | 2, 4 | syl5 34 | 1 ⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
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 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1782 |
This theorem is referenced by: ax12wdemo 2131 |
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