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Mirrors > Home > ILE Home > Th. List > exbid | GIF version |
Description: Formula-building rule for existential quantifier (deduction rule). (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 1452 | . 2 ⊢ (𝜑 → ∀𝑥𝜑) |
3 | exbid.2 | . 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
4 | 2, 3 | exbidh 1545 | 1 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 Ⅎwnf 1389 ∃wex 1421 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-4 1440 ax-ial 1467 |
This theorem depends on definitions: df-bi 115 df-nf 1390 |
This theorem is referenced by: mobid 1976 rexbida 2363 rexeqf 2546 opabbid 3843 repizf2 3936 oprabbid 5578 sscoll2 10783 |
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