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Theorem equsalv 2233
 Description: Version of equsal 2397 with a disjoint variable condition, which does not require ax-13 2346. See equsalvw 1991 for a version with two disjoint variable conditions requiring fewer axioms. See also the dual form equsexv 2234. (Contributed 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 2178 . 2 (∀𝑥(𝑥 = 𝑦𝜓) ↔ (∃𝑥 𝑥 = 𝑦𝜓))
3 equsalv.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
43pm5.74i 272 . . 3 ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑦𝜓))
54albii 1805 . 2 (∀𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜓))
6 ax6ev 1953 . . 3 𝑥 𝑥 = 𝑦
76a1bi 364 . 2 (𝜓 ↔ (∃𝑥 𝑥 = 𝑦𝜓))
82, 5, 73bitr4i 304 1 (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207  ∀wal 1523  ∃wex 1765  Ⅎwnf 1769 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-12 2143 This theorem depends on definitions:  df-bi 208  df-ex 1766  df-nf 1770 This theorem is referenced by:  equsalhw  2267  sbiev  2297  sb6rfv  2335  bj-equsalhv  33681
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