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Mirrors > Home > MPE Home > Th. List > ax12w | Structured version Visualization version GIF version |
Description: Weak version of ax-12 2163 from which we can prove any ax-12 2163 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 2129. (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 2084 | . 2 ⊢ (∀𝑦𝜑 → 𝜑) |
3 | ax12w.1 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 3 | ax12wlem 2126 | . 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 198 ∀wal 1599 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 |
This theorem depends on definitions: df-bi 199 df-an 387 df-ex 1824 |
This theorem is referenced by: ax12wdemo 2129 |
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