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Mirrors > Home > MPE Home > Th. List > ax12w | Structured version Visualization version GIF version |
Description: Weak version of ax-12 2166 from which we can prove any ax-12 2166 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 2126. (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 2033 | . 2 ⊢ (∀𝑦𝜑 → 𝜑) |
3 | ax12w.1 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 3 | ax12wlem 2123 | . 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 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 206 df-an 396 df-ex 1778 |
This theorem is referenced by: ax12wdemo 2126 |
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