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Mirrors > Home > NFE Home > Th. List > exanali | GIF version |
Description: A transformation of quantifiers and logical connectives. (Contributed by NM, 25-Mar-1996.) (Proof shortened by Wolf Lammen, 4-Sep-2014.) |
Ref | Expression |
---|---|
exanali | ⊢ (∃x(φ ∧ ¬ ψ) ↔ ¬ ∀x(φ → ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | annim 414 | . . 3 ⊢ ((φ ∧ ¬ ψ) ↔ ¬ (φ → ψ)) | |
2 | 1 | exbii 1582 | . 2 ⊢ (∃x(φ ∧ ¬ ψ) ↔ ∃x ¬ (φ → ψ)) |
3 | exnal 1574 | . 2 ⊢ (∃x ¬ (φ → ψ) ↔ ¬ ∀x(φ → ψ)) | |
4 | 2, 3 | bitri 240 | 1 ⊢ (∃x(φ ∧ ¬ ψ) ↔ ¬ ∀x(φ → ψ)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 176 ∧ wa 358 ∀wal 1540 ∃wex 1541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 |
This theorem depends on definitions: df-bi 177 df-an 360 df-ex 1542 |
This theorem is referenced by: ax11indn 2195 rexnal 2625 gencbval 2903 nss 3329 ssfin 4470 ncfinlowerlem1 4482 spfinex 4537 nfunv 5138 funsex 5828 fnfullfunlem1 5856 foundex 5914 fnfreclem1 6317 |
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