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Mirrors > Home > ILE Home > Th. List > exbidh | GIF version |
Description: Formula-building rule for existential quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
exbidh.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
exbidh.2 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
exbidh | ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exbidh.1 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | |
2 | exbidh.2 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
3 | 1, 2 | alrimih 1398 | . 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒)) |
4 | exbi 1535 | . 2 ⊢ (∀𝑥(𝜓 ↔ 𝜒) → (∃𝑥𝜓 ↔ ∃𝑥𝜒)) | |
5 | 3, 4 | syl 14 | 1 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∀wal 1282 ∃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 |
This theorem is referenced by: exbid 1547 drex2 1660 drex1 1719 exbidv 1746 mobidh 1975 |
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