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Theorem sbh 1706
Description: Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 17-Oct-2004.)
Hypothesis
Ref Expression
sbh.1 (𝜑 → ∀𝑥𝜑)
Assertion
Ref Expression
sbh ([𝑦 / 𝑥]𝜑𝜑)

Proof of Theorem sbh
StepHypRef Expression
1 sb1 1696 . . . 4 ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑))
2 sbh.1 . . . . 5 (𝜑 → ∀𝑥𝜑)
3219.41h 1620 . . . 4 (∃𝑥(𝑥 = 𝑦𝜑) ↔ (∃𝑥 𝑥 = 𝑦𝜑))
41, 3sylib 120 . . 3 ([𝑦 / 𝑥]𝜑 → (∃𝑥 𝑥 = 𝑦𝜑))
54simprd 112 . 2 ([𝑦 / 𝑥]𝜑𝜑)
6 stdpc4 1705 . . 3 (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
72, 6syl 14 . 2 (𝜑 → [𝑦 / 𝑥]𝜑)
85, 7impbii 124 1 ([𝑦 / 𝑥]𝜑𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1287  wex 1426  [wsb 1692
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-i9 1468  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-sb 1693
This theorem is referenced by:  sbf  1707  sb6x  1709  nfs1f  1710  hbs1f  1711  sbid2h  1777  sblimv  1822  sbrim  1878  sbrbif  1884  elsb3  1900  elsb4  1901
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