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Theorem sbh 1675
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 1665 . . . 4 ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑))
2 sbh.1 . . . . 5 (𝜑 → ∀𝑥𝜑)
3219.41h 1591 . . . 4 (∃𝑥(𝑥 = 𝑦𝜑) ↔ (∃𝑥 𝑥 = 𝑦𝜑))
41, 3sylib 131 . . 3 ([𝑦 / 𝑥]𝜑 → (∃𝑥 𝑥 = 𝑦𝜑))
54simprd 111 . 2 ([𝑦 / 𝑥]𝜑𝜑)
6 stdpc4 1674 . . 3 (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
72, 6syl 14 . 2 (𝜑 → [𝑦 / 𝑥]𝜑)
85, 7impbii 121 1 ([𝑦 / 𝑥]𝜑𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wal 1257  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-4 1416  ax-i9 1439  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-sb 1662
This theorem is referenced by:  sbf  1676  sb6x  1678  nfs1f  1679  hbs1f  1680  sbid2h  1745  sblimv  1790  sbrim  1846  sbrbif  1852  elsb3  1868  elsb4  1869
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