Theorem sbceq1a 3042

Description: Equality theorem for class substitution. Class version of sbequ12 1819. (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 1822	. 2 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑)
2		dfsbcq2 3035	. 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 1810  [wsbc 3032

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 2213

This theorem depends on definitions:  df-bi 117  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-sbc 3033

This theorem is referenced by:  sbceq2a  3043  elrabsf  3071  cbvralcsf  3191  cbvrexcsf  3192  rabsnifsb  3741  euotd  4353  omsinds  4726  elfvmptrab1  5750  ralrnmpt  5797  rexrnmpt  5798  riotass2  6010  riotass  6011  elovmporab  6232  elovmporab1w  6233  uchoice  6309  sbcopeq1a  6359  mpoxopoveq  6449  findcard2  7121  findcard2s  7122  ac6sfi  7130  opabfi  7175  dcfi  7223  indpi  7605  nn0ind-raph  9641  indstr  9871  fzrevral  10385  exfzdc  10532  zsupcllemstep  10535  infssuzex  10539  uzsinds  10752  wrdind  11352  wrd2ind  11353  prmind2  12755  gropd  15971  grstructd2dom  15972  bj-intabssel  16490  bj-bdfindes  16648  bj-findes  16680