Theorem sbceq1a 3052

Description: Equality theorem for class substitution. Class version of sbequ12 1820. (Contributed by NM, 26-Sep-2003.)

Assertion

Ref	Expression
sbceq1a	⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑))

Proof of Theorem sbceq1a

Step	Hyp	Ref	Expression
1		sbid 1823	. 2 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑)
2		dfsbcq2 3045	. 2 ⊢ (𝑥 = 𝐴 → ([𝑥 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
3	1, 2	bitr3id 194	1 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1398  [wsb 1811  [wsbc 3042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-sbc 3043

This theorem is referenced by:  sbceq2a  3053  elrabsf  3081  cbvralcsf  3201  cbvrexcsf  3202  rabsnifsb  3757  euotd  4371  omsinds  4744  elfvmptrab1  5772  ralrnmpt  5819  rexrnmpt  5820  riotass2  6032  riotass  6033  elovmporab  6254  elovmporab1w  6255  uchoice  6331  sbcopeq1a  6381  mpoxopoveq  6471  findcard2  7146  findcard2s  7147  ac6sfi  7155  opabfi  7200  dcfi  7268  indpi  7657  nn0ind-raph  9695  indstr  9925  fzrevral  10439  exfzdc  10586  zsupcllemstep  10589  infssuzex  10593  uzsinds  10806  wrdind  11414  wrd2ind  11415  prmind2  12817  gropd  16042  grstructd2dom  16043  bj-intabssel  16561  bj-bdfindes  16719  bj-findes  16751