Theorem sbceq1a 3038

Description: Equality theorem for class substitution. Class version of sbequ12 1817. (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 1820	. 2 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑)
2		dfsbcq2 3031	. 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 1395  [wsb 1808  [wsbc 3028

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-sbc 3029

This theorem is referenced by:  sbceq2a  3039  elrabsf  3067  cbvralcsf  3187  cbvrexcsf  3188  euotd  4340  omsinds  4713  elfvmptrab1  5728  ralrnmpt  5776  rexrnmpt  5777  riotass2  5982  riotass  5983  elovmporab  6204  elovmporab1w  6205  uchoice  6281  sbcopeq1a  6331  mpoxopoveq  6384  findcard2  7047  findcard2s  7048  ac6sfi  7056  opabfi  7096  dcfi  7144  indpi  7525  nn0ind-raph  9560  indstr  9784  fzrevral  10297  exfzdc  10441  zsupcllemstep  10444  infssuzex  10448  uzsinds  10661  wrdind  11249  wrd2ind  11250  prmind2  12637  gropd  15842  grstructd2dom  15843  bj-intabssel  16111  bj-bdfindes  16270  bj-findes  16302