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Theorem bj-sbievw2 34163
Description: Lemma for substitution. (Contributed by BJ, 23-Jul-2023.)
Assertion
Ref Expression
bj-sbievw2 ([𝑦 / 𝑥](𝜓𝜑) → (𝜓 → [𝑦 / 𝑥]𝜑))
Distinct variable groups:   𝜓,𝑥   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)

Proof of Theorem bj-sbievw2
StepHypRef Expression
1 sb6 2087 . 2 ([𝑦 / 𝑥](𝜓𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → (𝜓𝜑)))
2 bj-sblem2 34160 . . 3 (∀𝑥(𝑥 = 𝑦 → (𝜓𝜑)) → ((∃𝑥 𝑥 = 𝑦𝜓) → ∀𝑥(𝑥 = 𝑦𝜑)))
3 jarr 106 . . . 4 (((∃𝑥 𝑥 = 𝑦𝜓) → ∀𝑥(𝑥 = 𝑦𝜑)) → (𝜓 → ∀𝑥(𝑥 = 𝑦𝜑)))
4 sb6 2087 . . . 4 ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
53, 4syl6ibr 254 . . 3 (((∃𝑥 𝑥 = 𝑦𝜓) → ∀𝑥(𝑥 = 𝑦𝜑)) → (𝜓 → [𝑦 / 𝑥]𝜑))
62, 5syl 17 . 2 (∀𝑥(𝑥 = 𝑦 → (𝜓𝜑)) → (𝜓 → [𝑦 / 𝑥]𝜑))
71, 6sylbi 219 1 ([𝑦 / 𝑥](𝜓𝜑) → (𝜓 → [𝑦 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1529  wex 1774  [wsb 2063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009
This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1775  df-sb 2064
This theorem is referenced by: (None)
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