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Theorem equsal 2436
Description: An equivalence related to implicit substitution. See equsalvw 2086 and equsalv 2255 for versions with dv conditions proved from fewer axioms. See also the dual form equsex 2437. (Contributed by NM, 2-Jun-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 5-Feb-2018.)
Hypotheses
Ref Expression
equsal.1 𝑥𝜓
equsal.2 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
equsal (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Proof of Theorem equsal
StepHypRef Expression
1 equsal.1 . . 3 𝑥𝜓
2119.23 2227 . 2 (∀𝑥(𝑥 = 𝑦𝜓) ↔ (∃𝑥 𝑥 = 𝑦𝜓))
3 equsal.2 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
43pm5.74i 260 . . 3 ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑦𝜓))
54albii 1896 . 2 (∀𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜓))
6 ax6e 2395 . . 3 𝑥 𝑥 = 𝑦
76a1bi 351 . 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 1630  wex 1853  wnf 1857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-12 2196  ax-13 2391
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1854  df-nf 1859
This theorem is referenced by:  equsex  2437  equsalh  2439  dvelimf  2474  sb6x  2521  sb6rf  2560
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