Theorem sbequ12r 2255

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 2254	. . 3 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ [𝑥 / 𝑦]𝜑))
2	1	bicomd 223	. 2 ⊢ (𝑦 = 𝑥 → ([𝑥 / 𝑦]𝜑 ↔ 𝜑))
3	2	equcoms 2021	1 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 206  [wsb 2067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-12 2180

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-sb 2068

This theorem is referenced by:  sbelx  2256  sbequ12a  2257  sbid  2258  sbcov  2259  sbid2vw  2262  sb6rfv  2357  sbbib  2361  sb5rf  2467  sb6rf  2468  2sb5rf  2472  2sb6rf  2473  abbib  2800  opeliunxp  5678  opeliun2xp  5679  isarep1  6565  findes  7825  axrepndlem1  10478  axrepndlem2  10479  nn0min  32795  esumcvg  34091  bj-sbidmOLD  36884  bj-gabima  36974  wl-nfs1t  37571  wl-sbid2ft  37579  wl-equsb4  37591  wl-ax11-lem5  37623  sbcalf  38154  sbcexf  38155