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Theorem bj-equsalv 32413
Description: Version of equsal 2290 with a dv condition, which does not require ax-13 2245. See equsalvw 1928 for a version with two dv conditions requiring fewer axioms. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-equsalv.nf 𝑥𝜓
bj-equsalv.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
bj-equsalv (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem bj-equsalv
StepHypRef Expression
1 bj-equsalv.nf . . 3 𝑥𝜓
2119.23 2078 . 2 (∀𝑥(𝑥 = 𝑦𝜓) ↔ (∃𝑥 𝑥 = 𝑦𝜓))
3 bj-equsalv.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
43pm5.74i 260 . . 3 ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑦𝜓))
54albii 1744 . 2 (∀𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜓))
6 ax6ev 1887 . . 3 𝑥 𝑥 = 𝑦
76a1bi 352 . 2 (𝜓 ↔ (∃𝑥 𝑥 = 𝑦𝜓))
82, 5, 73bitr4i 292 1 (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1478  wex 1701  wnf 1705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-12 2044
This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1702  df-nf 1707
This theorem is referenced by:  bj-equsalhv  32414
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