Theorem sbequ12r 2248

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 2247	. . 3 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ [𝑥 / 𝑦]𝜑))
2	1	bicomd 222	. 2 ⊢ (𝑦 = 𝑥 → ([𝑥 / 𝑦]𝜑 ↔ 𝜑))
3	2	equcoms 2024	1 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 205  [wsb 2068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-12 2173

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-sb 2069

This theorem is referenced by:  sbelx  2249  sbequ12a  2250  sbid  2251  sbid2vw  2254  sb6rfv  2355  sbbib  2359  sb5rf  2467  sb6rf  2468  2sb5rf  2472  2sb6rf  2473  abbi  2811  opeliunxp  5645  isarep1  6506  findes  7723  axrepndlem1  10279  axrepndlem2  10280  nn0min  31036  esumcvg  31954  bj-sbidmOLD  34961  bj-gabima  35055  wl-nfs1t  35623  wl-sb6rft  35630  wl-equsb4  35639  wl-ax11-lem5  35667  sbcalf  36199  sbcexf  36200  opeliun2xp  45556