Theorem sbequ12r 2244

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 2243	. . 3 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ [𝑥 / 𝑦]𝜑))
2	1	bicomd 222	. 2 ⊢ (𝑦 = 𝑥 → ([𝑥 / 𝑦]𝜑 ↔ 𝜑))
3	2	equcoms 2022	1 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 205  [wsb 2066

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 1912  ax-6 1970  ax-7 2010  ax-12 2170

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1781  df-sb 2067

This theorem is referenced by:  sbelx  2245  sbequ12a  2246  sbid  2247  sbid2vw  2250  sb6rfv  2354  sbbib  2358  sb5rf  2466  sb6rf  2467  2sb5rf  2471  2sb6rf  2472  abbi  2809  opeliunxp  5673  isarep1  6560  isarep1OLD  6561  findes  7796  axrepndlem1  10428  axrepndlem2  10429  nn0min  31269  esumcvg  32194  bj-sbidmOLD  35107  bj-gabima  35201  wl-nfs1t  35768  wl-sb6rft  35775  wl-equsb4  35784  wl-ax11-lem5  35812  sbcalf  36344  sbcexf  36345  opeliun2xp  45933