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Mirrors > Home > MPE Home > Th. List > chvar | Structured version Visualization version GIF version |
Description: Implicit substitution of 𝑦 for 𝑥 into a theorem. Usage of this theorem is discouraged because it depends on ax-13 2367. Use the weaker chvarfv 2229 if possible. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by Mario Carneiro, 3-Oct-2016.) (New usage is discouraged.) |
Ref | Expression |
---|---|
chvar.1 | ⊢ Ⅎ𝑥𝜓 |
chvar.2 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
chvar.3 | ⊢ 𝜑 |
Ref | Expression |
---|---|
chvar | ⊢ 𝜓 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | chvar.1 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
2 | chvar.2 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
3 | 2 | biimpd 228 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
4 | 1, 3 | spim 2382 | . 2 ⊢ (∀𝑥𝜑 → 𝜓) |
5 | chvar.3 | . 2 ⊢ 𝜑 | |
6 | 4, 5 | mpg 1792 | 1 ⊢ 𝜓 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 Ⅎwnf 1778 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-12 2167 ax-13 2367 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1775 df-nf 1779 |
This theorem is referenced by: chvarv 2391 zfcndrep 10637 |
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