Theorem sbequ12r 2259

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 2258	. . 3 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ [𝑥 / 𝑦]𝜑))
2	1	bicomd 223	. 2 ⊢ (𝑦 = 𝑥 → ([𝑥 / 𝑦]𝜑 ↔ 𝜑))
3	2	equcoms 2021	1 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 206  [wsb 2067

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 1911  ax-6 1968  ax-7 2009  ax-12 2184

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-sb 2068

This theorem is referenced by:  sbelx  2260  sbequ12a  2261  sbid  2262  sbcov  2263  sbid2vw  2266  sb6rfv  2361  sbbib  2365  sb5rf  2471  sb6rf  2472  2sb5rf  2476  2sb6rf  2477  abbib  2805  opeliunxp  5691  opeliun2xp  5692  isarep1  6581  findes  7842  axrepndlem1  10503  axrepndlem2  10504  nn0min  32901  esumcvg  34243  bj-sbidmOLD  37051  bj-gabima  37141  wl-nfs1t  37742  wl-sbid2ft  37750  wl-equsb4  37762  sbcalf  38315  sbcexf  38316