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Theorem bj-equs45fv 32877
Description: Version of equs45f 2378 with a dv condition, which does not require ax-13 2282. Note that the version of equs5 2379 with a dv condition is actually sb56 2188 (up to adding a superfluous antecedent). (Contributed by BJ, 11-Sep-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-equs45fv.1 𝑦𝜑
Assertion
Ref Expression
bj-equs45fv (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bj-equs45fv
StepHypRef Expression
1 bj-equs45fv.1 . . . . . 6 𝑦𝜑
21nf5ri 2103 . . . . 5 (𝜑 → ∀𝑦𝜑)
32anim2i 592 . . . 4 ((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦 ∧ ∀𝑦𝜑))
43eximi 1802 . . 3 (∃𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑))
5 equs5a 2376 . . 3 (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))
64, 5syl 17 . 2 (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))
7 equs4v 1976 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))
86, 7impbii 199 1 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1521  wex 1744  wnf 1748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087  ax-13 2282
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750
This theorem is referenced by: (None)
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