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Mirrors > Home > ILE Home > Th. List > sbieh | GIF version |
Description: Conversion of implicit substitution to explicit substitution. New proofs should use sbie 1791 instead. (Contributed by NM, 30-Jun-1994.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sbieh.1 | ⊢ (𝜓 → ∀𝑥𝜓) |
sbieh.2 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
sbieh | ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 | . 2 ⊢ (𝜑 → 𝜑) | |
2 | 1 | hbth 1463 | . . 3 ⊢ ((𝜑 → 𝜑) → ∀𝑥(𝜑 → 𝜑)) |
3 | sbieh.1 | . . . 4 ⊢ (𝜓 → ∀𝑥𝜓) | |
4 | 3 | a1i 9 | . . 3 ⊢ ((𝜑 → 𝜑) → (𝜓 → ∀𝑥𝜓)) |
5 | sbieh.2 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
6 | 5 | a1i 9 | . . 3 ⊢ ((𝜑 → 𝜑) → (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))) |
7 | 2, 4, 6 | sbiedh 1787 | . 2 ⊢ ((𝜑 → 𝜑) → ([𝑦 / 𝑥]𝜑 ↔ 𝜓)) |
8 | 1, 7 | ax-mp 5 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∀wal 1351 [wsb 1762 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-4 1510 ax-i9 1530 ax-ial 1534 |
This theorem depends on definitions: df-bi 117 df-sb 1763 |
This theorem is referenced by: sbie 1791 sbco2vlem 1944 equsb3lem 1950 sbco2yz 1963 dvelimf 2015 elsb1 2155 elsb2 2156 |
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