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Theorem bj-equsexval 34053
Description: Special case of equsexv 2271 proved from Tarski, ax-10 2146 (modal5) and hba1 2303 (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 578 . . 3 ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑦 ∧ ∀𝑥𝜓))
32exbii 1849 . 2 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∃𝑥(𝑥 = 𝑦 ∧ ∀𝑥𝜓))
4 ax6ev 1973 . . 3 𝑥 𝑥 = 𝑦
5 bj-19.41al 34052 . . 3 (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑥𝜓) ↔ (∃𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝜓))
64, 5mpbiran 708 . 2 (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑥𝜓) ↔ ∀𝑥𝜓)
73, 6bitri 278 1 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536  wex 1781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-10 2146  ax-12 2179
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786
This theorem is referenced by: (None)
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