Theorem sbequ12r 2150

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 2149	. . 3 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ [𝑥 / 𝑦]𝜑))
2	1	bicomd 213	. 2 ⊢ (𝑦 = 𝑥 → ([𝑥 / 𝑦]𝜑 ↔ 𝜑))
3	2	equcoms 1993	1 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 196  [wsb 1937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-12 2087

This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-sb 1938

This theorem is referenced by:  sbequ12a  2151  sbid  2152  sb5rf  2450  sb6rf  2451  2sb5rf  2479  2sb6rf  2480  opeliunxp  5204  isarep1  6015  findes  7138  axrepndlem1  9452  axrepndlem2  9453  nn0min  29695  esumcvg  30276  bj-abbi  32900  bj-sbidmOLD  32956  wl-nfs1t  33454  wl-sb6rft  33460  wl-equsb4  33468  wl-ax11-lem5  33496  sbcalf  34047  sbcexf  34048  opeliun2xp  42436