Theorem sbequ12r 2253

Description: An equality theorem for substitution. (Contributed by NM, 6-Oct-2004.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)

Assertion

Ref	Expression
sbequ12r	⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 ↔ 𝜑))

Proof of Theorem sbequ12r

Step	Hyp	Ref	Expression
1		sbequ12 2252	. . 3 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ [𝑥 / 𝑦]𝜑))
2	1	bicomd 223	. 2 ⊢ (𝑦 = 𝑥 → ([𝑥 / 𝑦]𝜑 ↔ 𝜑))
3	2	equcoms 2020	1 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 206  [wsb 2065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2066

This theorem is referenced by:  sbelx  2254  sbequ12a  2255  sbid  2256  sbcov  2257  sbid2vw  2260  sb6rfv  2356  sbbib  2360  sb5rf  2466  sb6rf  2467  2sb5rf  2471  2sb6rf  2472  abbib  2799  opeliunxp  5708  opeliun2xp  5709  isarep1  6609  isarep1OLD  6610  findes  7879  axrepndlem1  10552  axrepndlem2  10553  nn0min  32752  esumcvg  34083  bj-sbidmOLD  36845  bj-gabima  36935  wl-nfs1t  37532  wl-sbid2ft  37540  wl-equsb4  37552  wl-ax11-lem5  37584  sbcalf  38115  sbcexf  38116