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Mirrors > Home > ILE Home > Th. List > hbsb2a | GIF version |
Description: Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.) |
Ref | Expression |
---|---|
hbsb2a | ⊢ ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb4a 1801 | . 2 ⊢ ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) | |
2 | sb2 1767 | . . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑) | |
3 | 2 | a5i 1543 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∀𝑥[𝑦 / 𝑥]𝜑) |
4 | 1, 3 | syl 14 | 1 ⊢ ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀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-11 1506 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: hbsb3 1808 |
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