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Theorem sbhypf 3357
Description: Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf 3658. (Contributed by Raph Levien, 10-Apr-2004.)
Hypotheses
Ref Expression
sbhypf.1 𝑥𝜓
sbhypf.2 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
sbhypf (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑𝜓))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐴(𝑦)

Proof of Theorem sbhypf
StepHypRef Expression
1 eqeq1 2728 . . 3 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 = 𝐴))
21equsexvw 2051 . 2 (∃𝑥(𝑥 = 𝑦𝑥 = 𝐴) ↔ 𝑦 = 𝐴)
3 nfs1v 2538 . . . 4 𝑥[𝑦 / 𝑥]𝜑
4 sbhypf.1 . . . 4 𝑥𝜓
53, 4nfbi 1946 . . 3 𝑥([𝑦 / 𝑥]𝜑𝜓)
6 sbequ12 2222 . . . . 5 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
76bicomd 213 . . . 4 (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑𝜑))
8 sbhypf.2 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
97, 8sylan9bb 738 . . 3 ((𝑥 = 𝑦𝑥 = 𝐴) → ([𝑦 / 𝑥]𝜑𝜓))
105, 9exlimi 2197 . 2 (∃𝑥(𝑥 = 𝑦𝑥 = 𝐴) → ([𝑦 / 𝑥]𝜑𝜓))
112, 10sylbir 225 1 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1596  wex 1817  wnf 1821  [wsb 2010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-12 2160  ax-13 2355  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-cleq 2717
This theorem is referenced by:  mob2  3492  reu2eqd  3509  cbvmptf  4856  ralxpf  5376  tfisi  7175  ac6sf  9424  nn0ind-raph  11590  ac6sf2  29659  nn0min  29797  ac6gf  33759  fdc1  33774
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