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Mirrors > Home > ILE Home > Th. List > sb4a | GIF version |
Description: A version of sb4 1805 that doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) |
Ref | Expression |
---|---|
sb4a | ⊢ ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb1 1740 | . 2 ⊢ ([𝑦 / 𝑥]∀𝑦𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑)) | |
2 | equs5a 1767 | . 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)) | |
3 | 1, 2 | syl 14 | 1 ⊢ ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wal 1330 ∃wex 1469 [wsb 1736 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-gen 1426 ax-ie2 1471 ax-11 1485 ax-ial 1515 |
This theorem depends on definitions: df-bi 116 df-sb 1737 |
This theorem is referenced by: sb6f 1776 hbsb2a 1779 |
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