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  2355  sbbib  2359  sb5rf  2465  sb6rf  2466  2sb5rf  2470  2sb6rf  2471  abbib  2798  opeliunxp  5690  opeliun2xp  5691  isarep1  6575  findes  7840  axrepndlem1  10505  axrepndlem2  10506  nn0min  32778  esumcvg  34055  bj-sbidmOLD  36826  bj-gabima  36916  wl-nfs1t  37513  wl-sbid2ft  37521  wl-equsb4  37533  wl-ax11-lem5  37565  sbcalf  38096  sbcexf  38097