Theorem sbequ12r 2252

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 2251	. . 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 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-12 2177

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2065

This theorem is referenced by:  sbelx  2253  sbequ12a  2254  sbid  2255  sbcov  2256  sbid2vw  2259  sb6rfv  2360  sbbib  2364  sb5rf  2472  sb6rf  2473  2sb5rf  2477  2sb6rf  2478  abbib  2811  opeliunxp  5752  opeliun2xp  5753  isarep1  6656  isarep1OLD  6657  findes  7922  axrepndlem1  10632  axrepndlem2  10633  nn0min  32822  esumcvg  34087  bj-sbidmOLD  36851  bj-gabima  36941  wl-nfs1t  37538  wl-sbid2ft  37546  wl-equsb4  37558  wl-ax11-lem5  37590  sbcalf  38121  sbcexf  38122