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Theorem bj-eqs 31850
 Description: A lemma for substitutions, proved from Tarski's FOL. The version without DV(𝑥, 𝑦) is true but requires ax-13 2234. 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 1842 . 2 (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
3 exim 1751 . . 3 (∀𝑥(𝑥 = 𝑦𝜑) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑))
4 ax6ev 1877 . . . 4 𝑥 𝑥 = 𝑦
5 pm2.27 41 . . . 4 (∃𝑥 𝑥 = 𝑦 → ((∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑) → ∃𝑥𝜑))
64, 5ax-mp 5 . . 3 ((∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑) → ∃𝑥𝜑)
7 ax5e 1829 . . 3 (∃𝑥𝜑𝜑)
83, 6, 73syl 18 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → 𝜑)
92, 8impbii 198 1 (𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473  ∃wex 1695 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875 This theorem depends on definitions:  df-bi 196  df-ex 1696 This theorem is referenced by:  bj-sb  31864
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