Theorem rexanali 3116

Description: A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.) (Proof shortened by Wolf Lammen, 27-Dec-2019.)

Assertion

Ref	Expression
rexanali	⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜓) ↔ ¬ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓))

Proof of Theorem rexanali

Step	Hyp	Ref	Expression
1		dfrex2 3089	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜓) ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ (𝜑 ∧ ¬ 𝜓))
2		iman 407	. . 3 ⊢ ((𝜑 → 𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓))
3	2	ralbii 3108	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥 ∈ 𝐴 ¬ (𝜑 ∧ ¬ 𝜓))
4	1, 3	xchbinxr 338	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜓) ↔ ¬ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ∀wral 3076  ∃wrex 3086

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-ral 3077  df-rex 3087

This theorem is used by:  nrexralim  3146  ceqsralbv  3611  frpoind  6340  frind  9732  qsqueeze  13253  ncoprmgcdne1b  16740  matunitlindflem1  22901  elcls  23298  ist1-2  23572  haust1  23577  t1sep  23595  bwth  23635  1stccnp  23688  filufint  24146  fclscf  24251  pmltpc  25678  ovolgelb  25708  itg2seq  25970  radcnvlt1  26654  pntlem3  27845  nosupbnd1lem5  27948  noinfbnd1lem5  27963  oncutlt  28529  umgr2edg1  29671  umgr2edgneu  29674  archiabl  33638  extdgfialglem1  34202  ordtconnlem1  34434  limsucncmpi  37064  ftc1anclem5  38446  clsk3nimkb  44880