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Theorem bj-sblem 34564
 Description: Lemma for substitution. (Contributed by BJ, 23-Jul-2023.)
Assertion
Ref Expression
bj-sblem (∀𝑥(𝜑 → (𝜓𝜒)) → (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜒)))
Distinct variable group:   𝜒,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem bj-sblem
StepHypRef Expression
1 pm5.74 273 . . . 4 ((𝜑 → (𝜓𝜒)) ↔ ((𝜑𝜓) ↔ (𝜑𝜒)))
21albii 1821 . . 3 (∀𝑥(𝜑 → (𝜓𝜒)) ↔ ∀𝑥((𝜑𝜓) ↔ (𝜑𝜒)))
3 albi 1820 . . 3 (∀𝑥((𝜑𝜓) ↔ (𝜑𝜒)) → (∀𝑥(𝜑𝜓) ↔ ∀𝑥(𝜑𝜒)))
42, 3sylbi 220 . 2 (∀𝑥(𝜑 → (𝜓𝜒)) → (∀𝑥(𝜑𝜓) ↔ ∀𝑥(𝜑𝜒)))
5 19.23v 1943 . 2 (∀𝑥(𝜑𝜒) ↔ (∃𝑥𝜑𝜒))
64, 5bitrdi 290 1 (∀𝑥(𝜑 → (𝜓𝜒)) → (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜒)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536  ∃wex 1781 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911 This theorem depends on definitions:  df-bi 210  df-ex 1782 This theorem is referenced by:  bj-sbievw  34567
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