ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  19.21h GIF version

Theorem 19.21h 1537
Description: Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑." New proofs should use 19.21 1563 instead. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
Hypothesis
Ref Expression
19.21h.1 (𝜑 → ∀𝑥𝜑)
Assertion
Ref Expression
19.21h (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))

Proof of Theorem 19.21h
StepHypRef Expression
1 19.21h.1 . . 3 (𝜑 → ∀𝑥𝜑)
2 alim 1434 . . 3 (∀𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
31, 2syl5 32 . 2 (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓))
4 hba1 1521 . . . 4 (∀𝑥𝜓 → ∀𝑥𝑥𝜓)
51, 4hbim 1525 . . 3 ((𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → ∀𝑥𝜓))
6 ax-4 1488 . . . 4 (∀𝑥𝜓𝜓)
76imim2i 12 . . 3 ((𝜑 → ∀𝑥𝜓) → (𝜑𝜓))
85, 7alrimih 1446 . 2 ((𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
93, 8impbii 125 1 (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-4 1488  ax-ial 1515  ax-i5r 1516
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  hbim1  1550  nf3  1648  19.21v  1846
  Copyright terms: Public domain W3C validator