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  5691  opeliun2xp  5692  isarep1  6581  findes  7844  axrepndlem1  10506  axrepndlem2  10507  nn0min  32909  esumcvg  34246  bj-sbidmOLD  37173  bj-gabima  37263  bj-axseprep  37397  wl-nfs1t  37876  wl-sbid2ft  37884  wl-equsb4  37896  sbcalf  38449  sbcexf  38450