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Theorem equsalv 2265
Description: An equivalence related to implicit substitution. Version of equsal 2428 with a disjoint variable condition, which does not require ax-13 2379. See equsalvw 2010 for a version with two disjoint variable conditions requiring fewer axioms. See also the dual form equsexv 2266. (Contributed by NM, 2-Jun-1993.) (Revised by BJ, 31-May-2019.)
Hypotheses
Ref Expression
equsalv.nf 𝑥𝜓
equsalv.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
equsalv (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem equsalv
StepHypRef Expression
1 equsalv.nf . . 3 𝑥𝜓
2119.23 2209 . 2 (∀𝑥(𝑥 = 𝑦𝜓) ↔ (∃𝑥 𝑥 = 𝑦𝜓))
3 equsalv.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
43pm5.74i 274 . . 3 ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑦𝜓))
54albii 1821 . 2 (∀𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜓))
6 ax6ev 1972 . . 3 𝑥 𝑥 = 𝑦
76a1bi 366 . 2 (𝜓 ↔ (∃𝑥 𝑥 = 𝑦𝜓))
82, 5, 73bitr4i 306 1 (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1536  wex 1781  wnf 1785
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 1911  ax-6 1970  ax-7 2015  ax-12 2175
This theorem depends on definitions:  df-bi 210  df-ex 1782  df-nf 1786
This theorem is referenced by:  equsalhw  2295  sbiev  2322  sb6rfv  2365  nfabdw  2976  bj-equsalhv  34243  ichnfimlem  43978
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