Theorem sbequ 1721

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, 5-Aug-1993.)

Assertion

Ref	Expression
sbequ	⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑))

Proof of Theorem sbequ

Step	Hyp	Ref	Expression
1		sbequi 1720	. 2 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))
2		sbequi 1720	. . 3 ⊢ (𝑦 = 𝑥 → ([𝑦 / 𝑧]𝜑 → [𝑥 / 𝑧]𝜑))
3	2	equcoms 1594	. 2 ⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑧]𝜑 → [𝑥 / 𝑧]𝜑))
4	1, 3	impbid 120	1 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑))

Syntax hints:   → wi 4   ↔ wb 98  [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646

This theorem is referenced by:  drsb2  1722  sbco2vlem  1820  sbco2yz  1837  sbcocom  1844  sb10f  1871  hbsb4  1888  nfsb4or  1899  sb8eu  1913  sb8euh  1923  cbvab  2160  cbvralf  2527  cbvrexf  2528  cbvreu  2531  cbvralsv  2544  cbvrexsv  2545  cbvrab  2555  cbvreucsf  2910  cbvrabcsf  2911  sbss  3329  cbvopab1  3830  cbvmpt  3851  tfis  4306  findes  4326  cbviota  4872  sb8iota  4874  cbvriota  5478  uzind4s  8533  cbvrald  9927  setindft  10090