Theorem sbequ 1779

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 1778	. 2 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))
2		sbequi 1778	. . 3 ⊢ (𝑦 = 𝑥 → ([𝑦 / 𝑧]𝜑 → [𝑥 / 𝑧]𝜑))
3	2	equcoms 1652	. 2 ⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑧]𝜑 → [𝑥 / 𝑧]𝜑))
4	1, 3	impbid 128	1 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑))

Syntax hints:   → wi 4   ↔ wb 104  [wsb 1703

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483

This theorem depends on definitions:  df-bi 116  df-nf 1405  df-sb 1704

This theorem is referenced by:  drsb2  1780  sbco2vlem  1880  sbco2yz  1897  sbcocom  1904  sb10f  1931  hbsb4  1948  nfsb4or  1959  sb8eu  1973  sb8euh  1983  cbvab  2222  cbvralf  2606  cbvrexf  2607  cbvreu  2610  cbvralsv  2623  cbvrexsv  2624  cbvrab  2639  cbvreucsf  3014  cbvrabcsf  3015  sbss  3418  disjiun  3870  cbvopab1  3941  cbvmpt  3963  tfis  4435  findes  4455  cbviota  5029  sb8iota  5031  cbvriota  5672  uzind4s  9235  bezoutlemmain  11479  cbvrald  12576  setindft  12748