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Mirrors > Home > ILE Home > Th. List > chvar | GIF version |
Description: Implicit substitution of 𝑦 for 𝑥 into a theorem. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by Mario Carneiro, 3-Oct-2016.) |
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 143 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
4 | 1, 3 | spim 1674 | . 2 ⊢ (∀𝑥𝜑 → 𝜓) |
5 | chvar.3 | . 2 ⊢ 𝜑 | |
6 | 4, 5 | mpg 1386 | 1 ⊢ 𝜓 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 Ⅎwnf 1395 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1382 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-4 1446 ax-i9 1469 ax-ial 1473 |
This theorem depends on definitions: df-bi 116 df-nf 1396 |
This theorem is referenced by: csbhypf 2969 opelopabsb 4098 findes 4433 fvmptssdm 5402 dfoprab4f 5979 dom2lem 6545 uzind4s 9141 fsumsplitf 10865 |
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