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Mirrors > Home > ILE Home > Th. List > equs5a | GIF version |
Description: A property related to substitution that unlike equs5 1822 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) |
Ref | Expression |
---|---|
equs5a | ⊢ (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hba1 1533 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∀𝑥∀𝑥(𝑥 = 𝑦 → 𝜑)) | |
2 | ax-11 1499 | . . 3 ⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
3 | 2 | imp 123 | . 2 ⊢ ((𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
4 | 1, 3 | exlimih 1586 | 1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wal 1346 ∃wex 1485 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-gen 1442 ax-ie2 1487 ax-11 1499 ax-ial 1527 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: equs5e 1788 sb4a 1794 equs45f 1795 |
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