Theorem sbequ12r 2098

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 2097	. . 3 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ [𝑥 / 𝑦]𝜑))
2	1	bicomd 212	. 2 ⊢ (𝑦 = 𝑥 → ([𝑥 / 𝑦]𝜑 ↔ 𝜑))
3	2	equcoms 1934	1 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 195  [wsb 1867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-sb 1868

This theorem is referenced by:  sbequ12a  2099  sbid  2100  sb5rf  2410  sb6rf  2411  2sb5rf  2439  2sb6rf  2440  opeliunxp  5093  isarep1  5891  findes  6988  axrepndlem1  9293  axrepndlem2  9294  nn0min  28954  esumcvg  29475  bj-abbi  31963  bj-sbidmOLD  32021  wl-nfs1t  32503  wl-sb6rft  32509  wl-equsb4  32517  wl-ax11-lem5  32545  sbcalf  33087  sbcexf  33088  opeliun2xp  41904