Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-equsalv Structured version   Visualization version   GIF version

Theorem bj-equsalv 31931
 Description: Version of equsal 2279 with a dv condition, which does not require ax-13 2234. See equsalvw 1918 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 2067 . 2 (∀𝑥(𝑥 = 𝑦𝜓) ↔ (∃𝑥 𝑥 = 𝑦𝜓))
3 bj-equsalv.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
43pm5.74i 259 . . 3 ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑦𝜓))
54albii 1737 . 2 (∀𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜓))
6 ax6ev 1877 . . 3 𝑥 𝑥 = 𝑦
76a1bi 351 . 2 (𝜓 ↔ (∃𝑥 𝑥 = 𝑦𝜓))
82, 5, 73bitr4i 291 1 (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473  ∃wex 1695  Ⅎwnf 1699 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-or 384  df-ex 1696  df-nf 1701 This theorem is referenced by:  bj-equsalhv  31932
 Copyright terms: Public domain W3C validator