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Theorem bj-sbievw 34230
 Description: Lemma for substitution. Closed form of equsalvw 2011 and sbievw 2104. (Contributed by BJ, 23-Jul-2023.)
Assertion
Ref Expression
bj-sbievw ([𝑦 / 𝑥](𝜑𝜓) → ([𝑦 / 𝑥]𝜑𝜓))
Distinct variable groups:   𝜓,𝑥   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)

Proof of Theorem bj-sbievw
StepHypRef Expression
1 sb6 2094 . 2 ([𝑦 / 𝑥](𝜑𝜓) ↔ ∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)))
2 bj-sblem 34227 . . 3 (∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → (∀𝑥(𝑥 = 𝑦𝜑) ↔ (∃𝑥 𝑥 = 𝑦𝜓)))
3 sb6 2094 . . 3 ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
4 ax6ev 1973 . . . 4 𝑥 𝑥 = 𝑦
54a1bi 366 . . 3 (𝜓 ↔ (∃𝑥 𝑥 = 𝑦𝜓))
62, 3, 53bitr4g 317 . 2 (∀𝑥(𝑥 = 𝑦 → (𝜑𝜓)) → ([𝑦 / 𝑥]𝜑𝜓))
71, 6sylbi 220 1 ([𝑦 / 𝑥](𝜑𝜓) → ([𝑦 / 𝑥]𝜑𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536  ∃wex 1781  [wsb 2070 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 1912  ax-6 1971  ax-7 2016 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071 This theorem is referenced by: (None)
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