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Mirrors > Home > ILE Home > Th. List > exbi | GIF version |
Description: Theorem 19.18 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
exbi | ⊢ (∀x(φ ↔ ψ) → (∃xφ ↔ ∃xψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bi1 111 | . . . 4 ⊢ ((φ ↔ ψ) → (φ → ψ)) | |
2 | 1 | alimi 1341 | . . 3 ⊢ (∀x(φ ↔ ψ) → ∀x(φ → ψ)) |
3 | exim 1487 | . . 3 ⊢ (∀x(φ → ψ) → (∃xφ → ∃xψ)) | |
4 | 2, 3 | syl 14 | . 2 ⊢ (∀x(φ ↔ ψ) → (∃xφ → ∃xψ)) |
5 | bi2 121 | . . . 4 ⊢ ((φ ↔ ψ) → (ψ → φ)) | |
6 | 5 | alimi 1341 | . . 3 ⊢ (∀x(φ ↔ ψ) → ∀x(ψ → φ)) |
7 | exim 1487 | . . 3 ⊢ (∀x(ψ → φ) → (∃xψ → ∃xφ)) | |
8 | 6, 7 | syl 14 | . 2 ⊢ (∀x(φ ↔ ψ) → (∃xψ → ∃xφ)) |
9 | 4, 8 | impbid 120 | 1 ⊢ (∀x(φ ↔ ψ) → (∃xφ ↔ ∃xψ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∀wal 1240 ∃wex 1378 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1333 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-4 1397 ax-ial 1424 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: exbii 1493 exbidh 1502 exintrbi 1521 19.19 1553 rexrnmpt 5253 |
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