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Theorem bj-equsexval 32622
 Description: Special case of equsexv 2108 proved from Tarski, ax-10 2018 (modal5) and hba1 2150 (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 669 . . 3 ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑦 ∧ ∀𝑥𝜓))
32exbii 1773 . 2 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∃𝑥(𝑥 = 𝑦 ∧ ∀𝑥𝜓))
4 ax6ev 1889 . . 3 𝑥 𝑥 = 𝑦
5 bj-19.41al 32621 . . 3 (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑥𝜓) ↔ (∃𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝜓))
64, 5mpbiran 953 . 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 384  ∀wal 1480  ∃wex 1703 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-10 2018  ax-12 2046 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1704  df-nf 1709 This theorem is referenced by: (None)
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