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Theorem sbequ 2404
Description: An equality theorem for substitution. Used in proof of Theorem 9.7 in [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 14-May-1993.)
Assertion
Ref Expression
sbequ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑))

Proof of Theorem sbequ
StepHypRef Expression
1 sbequi 2403 . 2 (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))
2 sbequi 2403 . . 3 (𝑦 = 𝑥 → ([𝑦 / 𝑧]𝜑 → [𝑥 / 𝑧]𝜑))
32equcoms 1993 . 2 (𝑥 = 𝑦 → ([𝑦 / 𝑧]𝜑 → [𝑥 / 𝑧]𝜑))
41, 3impbid 202 1 (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  [wsb 1937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087  ax-13 2282
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750  df-sb 1938
This theorem is referenced by:  drsb2  2406  sbcom3  2439  sbco2  2443  sbcom2  2473  sb10f  2484  sb8eu  2532  cbvralf  3195  cbvreu  3199  cbvralsv  3212  cbvrexsv  3213  cbvrab  3229  cbvreucsf  3600  cbvrabcsf  3601  sbss  4117  cbvopab1  4756  cbvmpt  4782  cbviota  5894  sb8iota  5896  cbvriota  6661  tfis  7096  tfinds  7101  findes  7138  uzind4s  11786  bj-cleljustab  32972  wl-sbcom2d-lem1  33472  wl-sb8eut  33489  wl-sbcom3  33502  sbeqi  34098  disjinfi  39694
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