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Theorem sbhypf 2730
Description: Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf . (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 vex 2684 . . 3 𝑦 ∈ V
2 eqeq1 2144 . . 3 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 = 𝐴))
31, 2ceqsexv 2720 . 2 (∃𝑥(𝑥 = 𝑦𝑥 = 𝐴) ↔ 𝑦 = 𝐴)
4 nfs1v 1910 . . . 4 𝑥[𝑦 / 𝑥]𝜑
5 sbhypf.1 . . . 4 𝑥𝜓
64, 5nfbi 1568 . . 3 𝑥([𝑦 / 𝑥]𝜑𝜓)
7 sbequ12 1744 . . . . 5 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
87bicomd 140 . . . 4 (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑𝜑))
9 sbhypf.2 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
108, 9sylan9bb 457 . . 3 ((𝑥 = 𝑦𝑥 = 𝐴) → ([𝑦 / 𝑥]𝜑𝜓))
116, 10exlimi 1573 . 2 (∃𝑥(𝑥 = 𝑦𝑥 = 𝐴) → ([𝑦 / 𝑥]𝜑𝜓))
123, 11sylbir 134 1 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wnf 1436  wex 1468  [wsb 1735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-v 2683
This theorem is referenced by:  mob2  2859  cbvmptf  4017  tfisi  4496  ralxpf  4680  rexxpf  4681  nn0ind-raph  9161
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