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Mirrors > Home > ILE Home > Th. List > Mathboxes > ch2var | GIF version |
Description: Implicit substitution of 𝑦 for 𝑥 and 𝑡 for 𝑧 into a theorem. (Contributed by BJ, 17-Oct-2019.) |
Ref | Expression |
---|---|
ch2var.nfx | ⊢ Ⅎ𝑥𝜓 |
ch2var.nfz | ⊢ Ⅎ𝑧𝜓 |
ch2var.maj | ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝜑 ↔ 𝜓)) |
ch2var.min | ⊢ 𝜑 |
Ref | Expression |
---|---|
ch2var | ⊢ 𝜓 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ch2var.nfx | . . 3 ⊢ Ⅎ𝑥𝜓 | |
2 | ch2var.nfz | . . 3 ⊢ Ⅎ𝑧𝜓 | |
3 | ch2var.maj | . . . 4 ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝜑 ↔ 𝜓)) | |
4 | 3 | biimpd 143 | . . 3 ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝜑 → 𝜓)) |
5 | 1, 2, 4 | 2spim 13801 | . 2 ⊢ (∀𝑧∀𝑥𝜑 → 𝜓) |
6 | ch2var.min | . . 3 ⊢ 𝜑 | |
7 | 6 | ax-gen 1442 | . 2 ⊢ ∀𝑥𝜑 |
8 | 5, 7 | mpg 1444 | 1 ⊢ 𝜓 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∀wal 1346 Ⅎwnf 1453 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1440 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 |
This theorem depends on definitions: df-bi 116 df-nf 1454 |
This theorem is referenced by: ch2varv 13803 |
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