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Mirrors > Home > ILE Home > Th. List > sb4a | GIF version |
Description: A version of sb4 1710 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 1646 | . 2 ⊢ ([y / x]∀yφ → ∃x(x = y ∧ ∀yφ)) | |
2 | equs5a 1672 | . 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 97 ∀wal 1240 ∃wex 1378 [wsb 1642 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-gen 1335 ax-ie2 1380 ax-11 1394 ax-ial 1424 |
This theorem depends on definitions: df-bi 110 df-sb 1643 |
This theorem is referenced by: sb6f 1681 hbsb2a 1684 |
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