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Theorem equsal 1612
Description: A useful equivalence related to substitution. (Contributed by NM, 5-Aug-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 1565 . 2 (x(x = yψ) ↔ (x x = yψ))
3 equsal.2 . . . 4 (x = y → (φψ))
43pm5.74i 169 . . 3 ((x = yφ) ↔ (x = yψ))
54albii 1356 . 2 (x(x = yφ) ↔ x(x = yψ))
6 a9e 1583 . . 3 x x = y
76a1bi 232 . 2 (ψ ↔ (x x = yψ))
82, 5, 73bitr4i 201 1 (x(x = yφ) ↔ ψ)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1240  wnf 1346  wex 1378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-i9 1420  ax-ial 1424  ax-i5r 1425
This theorem depends on definitions:  df-bi 110  df-nf 1347
This theorem is referenced by:  intirr  4654  fun11  4909
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