Theorem sbequ12r 2260

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 2259	. . 3 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ [𝑥 / 𝑦]𝜑))
2	1	bicomd 223	. 2 ⊢ (𝑦 = 𝑥 → ([𝑥 / 𝑦]𝜑 ↔ 𝜑))
3	2	equcoms 2022	1 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 206  [wsb 2068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-12 2185

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-sb 2069

This theorem is referenced by:  sbelx  2261  sbequ12a  2262  sbid  2263  sbcov  2264  sbid2vw  2267  sb6rfv  2361  sbbib  2365  sb5rf  2471  sb6rf  2472  2sb5rf  2476  2sb6rf  2477  abbib  2805  opeliunxp  5698  opeliun2xp  5699  isarep1  6587  findes  7851  axrepndlem1  10515  axrepndlem2  10516  nn0min  32894  esumcvg  34230  bj-sbidmOLD  37157  bj-gabima  37247  bj-axseprep  37381  wl-nfs1t  37862  wl-sbid2ft  37870  wl-equsb4  37882  sbcalf  38435  sbcexf  38436