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Mirrors > Home > ILE Home > Th. List > exbid | GIF version |
Description: Formula-building rule for existential quantifier (deduction form). (Contributed by Mario Carneiro, 24-Sep-2016.) |
Ref | Expression |
---|---|
exbid.1 | ⊢ Ⅎ𝑥𝜑 |
exbid.2 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
exbid | ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exbid.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | nfri 1499 | . 2 ⊢ (𝜑 → ∀𝑥𝜑) |
3 | exbid.2 | . 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
4 | 2, 3 | exbidh 1594 | 1 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 Ⅎwnf 1440 ∃wex 1472 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1427 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-4 1490 ax-ial 1514 |
This theorem depends on definitions: df-bi 116 df-nf 1441 |
This theorem is referenced by: mobid 2041 rexbida 2452 rexbid2 2462 rexeqf 2649 opabbid 4029 repizf2 4123 oprabbid 5874 |
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