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  2359  sbbib  2363  sb5rf  2471  sb6rf  2472  2sb5rf  2476  2sb6rf  2477  abbib  2804  opeliunxp  5721  opeliun2xp  5722  isarep1  6626  isarep1OLD  6627  findes  7896  axrepndlem1  10606  axrepndlem2  10607  nn0min  32799  esumcvg  34117  bj-sbidmOLD  36868  bj-gabima  36958  wl-nfs1t  37555  wl-sbid2ft  37563  wl-equsb4  37575  wl-ax11-lem5  37607  sbcalf  38138  sbcexf  38139