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  2362  sbbib  2366  sb5rf  2472  sb6rf  2473  2sb5rf  2477  2sb6rf  2478  abbib  2806  opeliunxp  5699  opeliun2xp  5700  isarep1  6589  findes  7852  axrepndlem1  10515  axrepndlem2  10516  nn0min  32911  esumcvg  34263  bj-sbidmOLD  37095  bj-gabima  37185  bj-axseprep  37319  wl-nfs1t  37789  wl-sbid2ft  37797  wl-equsb4  37809  sbcalf  38362  sbcexf  38363