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Theorem bj-eqs 32961
Description: A lemma for substitutions, proved from Tarski's FOL. The version without DV(𝑥, 𝑦) is true but requires ax-13 2383. The DV condition DV( 𝑥, 𝜑) is necessary for both directions: consider substituting 𝑥 = 𝑧 for 𝜑. (Contributed by BJ, 25-May-2021.)
Assertion
Ref Expression
bj-eqs (𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
Distinct variable groups:   𝑥,𝑦   𝜑,𝑥
Allowed substitution hint:   𝜑(𝑦)

Proof of Theorem bj-eqs
StepHypRef Expression
1 ax-1 6 . . 3 (𝜑 → (𝑥 = 𝑦𝜑))
21alrimiv 1996 . 2 (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
3 exim 1902 . . 3 (∀𝑥(𝑥 = 𝑦𝜑) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑))
4 ax6ev 2048 . . . 4 𝑥 𝑥 = 𝑦
5 pm2.27 42 . . . 4 (∃𝑥 𝑥 = 𝑦 → ((∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑) → ∃𝑥𝜑))
64, 5ax-mp 5 . . 3 ((∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑) → ∃𝑥𝜑)
7 ax5e 1982 . . 3 (∃𝑥𝜑𝜑)
83, 6, 73syl 18 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → 𝜑)
92, 8impbii 199 1 (𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1622  wex 1845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046
This theorem depends on definitions:  df-bi 197  df-ex 1846
This theorem is referenced by:  bj-sb  32975
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