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 2019	1 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 206  [wsb 2064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-sb 2065

This theorem is referenced by:  sbelx  2254  sbequ12a  2255  sbid  2256  sbcov  2257  sbid2vw  2260  sb6rfv  2363  sbbib  2367  sb5rf  2475  sb6rf  2476  2sb5rf  2480  2sb6rf  2481  abbib  2814  opeliunxp  5767  isarep1  6667  isarep1OLD  6668  findes  7940  axrepndlem1  10661  axrepndlem2  10662  nn0min  32824  esumcvg  34050  bj-sbidmOLD  36816  bj-gabima  36906  wl-nfs1t  37491  wl-sbid2ft  37499  wl-equsb4  37511  wl-ax11-lem5  37543  sbcalf  38074  sbcexf  38075  opeliun2xp  48057