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Theorem bj-equsexval 32763
Description: Special case of equsexv 2147 proved from Tarski, ax-10 2059 (modal5) and hba1 2189 (modal4). (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-equsexval.1 (𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥𝜓))
Assertion
Ref Expression
bj-equsexval (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem bj-equsexval
StepHypRef Expression
1 bj-equsexval.1 . . . 4 (𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥𝜓))
21pm5.32i 670 . . 3 ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑦 ∧ ∀𝑥𝜓))
32exbii 1814 . 2 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∃𝑥(𝑥 = 𝑦 ∧ ∀𝑥𝜓))
4 ax6ev 1947 . . 3 𝑥 𝑥 = 𝑦
5 bj-19.41al 32762 . . 3 (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑥𝜓) ↔ (∃𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝜓))
64, 5mpbiran 973 . 2 (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑥𝜓) ↔ ∀𝑥𝜓)
73, 6bitri 264 1 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1521  wex 1744
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
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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