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Theorem sbhypf 2786
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 2740 . . 3 𝑦 ∈ V
2 eqeq1 2184 . . 3 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 = 𝐴))
31, 2ceqsexv 2776 . 2 (∃𝑥(𝑥 = 𝑦𝑥 = 𝐴) ↔ 𝑦 = 𝐴)
4 nfs1v 1939 . . . 4 𝑥[𝑦 / 𝑥]𝜑
5 sbhypf.1 . . . 4 𝑥𝜓
64, 5nfbi 1589 . . 3 𝑥([𝑦 / 𝑥]𝜑𝜓)
7 sbequ12 1771 . . . . 5 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
87bicomd 141 . . . 4 (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑𝜑))
9 sbhypf.2 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
108, 9sylan9bb 462 . . 3 ((𝑥 = 𝑦𝑥 = 𝐴) → ([𝑦 / 𝑥]𝜑𝜓))
116, 10exlimi 1594 . 2 (∃𝑥(𝑥 = 𝑦𝑥 = 𝐴) → ([𝑦 / 𝑥]𝜑𝜓))
123, 11sylbir 135 1 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wnf 1460  wex 1492  [wsb 1762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-v 2739
This theorem is referenced by:  mob2  2917  cbvmptf  4097  tfisi  4586  ralxpf  4773  rexxpf  4774  nn0ind-raph  9369
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