exanali - New Foundations Explorer

	New Foundations Explorer		< Previous Next > Nearby theorems
Mirrors > Home > NFE Home > Th. List > exanali		GIF version

Theorem exanali 1585

Description: A transformation of quantifiers and logical connectives. (Contributed by NM, 25-Mar-1996.) (Proof shortened by Wolf Lammen, 4-Sep-2014.)

**Assertion**
Ref	Expression
exanali	⊢ (∃x(φ ∧ ¬ ψ) ↔ ¬ ∀x(φ → ψ))

**Proof of Theorem exanali**
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-1 5 ax-2 6 ax-3 7 ax-mp 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

W3C validator