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Theorem bj-sblem2 34052
Description: Lemma for substitution. (Contributed by BJ, 23-Jul-2023.)
Assertion
Ref Expression
bj-sblem2 (∀𝑥(𝜑 → (𝜒𝜓)) → ((∃𝑥𝜑𝜒) → ∀𝑥(𝜑𝜓)))
Distinct variable group:   𝜒,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem bj-sblem2
StepHypRef Expression
1 19.23v 1936 . 2 (∀𝑥(𝜑𝜒) ↔ (∃𝑥𝜑𝜒))
2 ax-2 7 . . 3 ((𝜑 → (𝜒𝜓)) → ((𝜑𝜒) → (𝜑𝜓)))
32al2imi 1809 . 2 (∀𝑥(𝜑 → (𝜒𝜓)) → (∀𝑥(𝜑𝜒) → ∀𝑥(𝜑𝜓)))
41, 3syl5bir 244 1 (∀𝑥(𝜑 → (𝜒𝜓)) → ((∃𝑥𝜑𝜒) → ∀𝑥(𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1528  wex 1773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904
This theorem depends on definitions:  df-bi 208  df-ex 1774
This theorem is referenced by:  bj-sbievw2  34055
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