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  4341  omsinds  4714  elfvmptrab1  5731  ralrnmpt  5779  rexrnmpt  5780  riotass2  5989  riotass  5990  elovmporab  6211  elovmporab1w  6212  uchoice  6289  sbcopeq1a  6339  mpoxopoveq  6392  findcard2  7059  findcard2s  7060  ac6sfi  7068  opabfi  7111  dcfi  7159  indpi  7540  nn0ind-raph  9575  indstr  9800  fzrevral  10313  exfzdc  10458  zsupcllemstep  10461  infssuzex  10465  uzsinds  10678  wrdind  11269  wrd2ind  11270  prmind2  12657  gropd  15863  grstructd2dom  15864  bj-intabssel  16208  bj-bdfindes  16367  bj-findes  16399