Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ralrexbid Structured version   Visualization version   GIF version

Theorem ralrexbid 3281
 Description: Formula-building rule for restricted existential quantifier, using a restricted universal quantifier to bind the quantified variable in the antecedent. (Contributed by AV, 21-Oct-2023.) Reduce axiom usage. (Revised by SN, 13-Nov-2023.)
Hypothesis
Ref Expression
ralrexbid.1 (𝜑 → (𝜓𝜃))
Assertion
Ref Expression
ralrexbid (∀𝑥𝐴 𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜃))

Proof of Theorem ralrexbid
StepHypRef Expression
1 df-ral 3111 . . 3 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
2 ralrexbid.1 . . . . . 6 (𝜑 → (𝜓𝜃))
32imim2i 16 . . . . 5 ((𝑥𝐴𝜑) → (𝑥𝐴 → (𝜓𝜃)))
43pm5.32d 580 . . . 4 ((𝑥𝐴𝜑) → ((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜃)))
54alexbii 1834 . . 3 (∀𝑥(𝑥𝐴𝜑) → (∃𝑥(𝑥𝐴𝜓) ↔ ∃𝑥(𝑥𝐴𝜃)))
61, 5sylbi 220 . 2 (∀𝑥𝐴 𝜑 → (∃𝑥(𝑥𝐴𝜓) ↔ ∃𝑥(𝑥𝐴𝜃)))
7 df-rex 3112 . 2 (∃𝑥𝐴 𝜓 ↔ ∃𝑥(𝑥𝐴𝜓))
8 df-rex 3112 . 2 (∃𝑥𝐴 𝜃 ↔ ∃𝑥(𝑥𝐴𝜃))
96, 7, 83bitr4g 317 1 (∀𝑥𝐴 𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜃))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃wex 1781   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-ral 3111  df-rex 3112 This theorem is referenced by:  dmopab2rex  5755  fiun  7636  f1iun  7637  dmopab3rexdif  32828
 Copyright terms: Public domain W3C validator