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Theorem sbhypf 2620
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 2577 . . 3 𝑦 ∈ V
2 eqeq1 2062 . . 3 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 = 𝐴))
31, 2ceqsexv 2610 . 2 (∃𝑥(𝑥 = 𝑦𝑥 = 𝐴) ↔ 𝑦 = 𝐴)
4 nfs1v 1831 . . . 4 𝑥[𝑦 / 𝑥]𝜑
5 sbhypf.1 . . . 4 𝑥𝜓
64, 5nfbi 1497 . . 3 𝑥([𝑦 / 𝑥]𝜑𝜓)
7 sbequ12 1670 . . . . 5 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
87bicomd 133 . . . 4 (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑𝜑))
9 sbhypf.2 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
108, 9sylan9bb 443 . . 3 ((𝑥 = 𝑦𝑥 = 𝐴) → ([𝑦 / 𝑥]𝜑𝜓))
116, 10exlimi 1501 . 2 (∃𝑥(𝑥 = 𝑦𝑥 = 𝐴) → ([𝑦 / 𝑥]𝜑𝜓))
123, 11sylbir 129 1 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wnf 1365  wex 1397  [wsb 1661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-v 2576
This theorem is referenced by:  mob2  2743  tfisi  4337  ralxpf  4509  rexxpf  4510  nn0ind-raph  8413
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