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Mirrors > Home > MPE Home > Th. List > ax12w | Structured version Visualization version GIF version |
Description: Weak version of ax-12 2047 from which we can prove any ax-12 2047 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 2012. (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 1967 | . 2 ⊢ (∀𝑦𝜑 → 𝜑) |
3 | ax12w.1 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 3 | ax12wlem 2009 | . 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 196 ∀wal 1481 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 |
This theorem is referenced by: ax12wdemo 2012 |
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