Theorem sbequ 1718

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 1717	. 2 ⊢ (x = y → ([x / z]φ → [y / z]φ))
2		sbequi 1717	. . 3 ⊢ (y = x → ([y / z]φ → [x / z]φ))
3	2	equcoms 1591	. 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 1642

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643

This theorem is referenced by:  drsb2  1719  sbco2vlem  1817  sbco2yz  1834  sbcocom  1841  sb10f  1868  hbsb4  1885  nfsb4or  1896  sb8eu  1910  sb8euh  1920  cbvab  2157  cbvralf  2521  cbvrexf  2522  cbvreu  2525  cbvralsv  2538  cbvrexsv  2539  cbvrab  2549  cbvreucsf  2904  cbvrabcsf  2905  sbss  3323  cbvopab1  3821  cbvmpt  3842  tfis  4249  findes  4269  cbviota  4815  sb8iota  4817  cbvriota  5421  uzind4s  8269  cbvrald  9196  setindft  9349