Theorem sbceq1a 3054

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 3047	. 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 3044

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 2216

This theorem depends on definitions:  df-bi 117  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-sbc 3045

This theorem is referenced by:  sbceq2a  3055  elrabsf  3083  cbvralcsf  3203  cbvrexcsf  3204  rabsnifsb  3759  euotd  4373  omsinds  4746  elfvmptrab1  5774  ralrnmpt  5821  rexrnmpt  5822  riotass2  6034  riotass  6035  elovmporab  6256  elovmporab1w  6257  uchoice  6333  sbcopeq1a  6383  mpoxopoveq  6473  findcard2  7148  findcard2s  7149  ac6sfi  7157  opabfi  7202  dcfi  7270  indpi  7659  nn0ind-raph  9698  indstr  9928  fzrevral  10443  exfzdc  10590  zsupcllemstep  10593  infssuzex  10597  uzsinds  10810  wrdind  11418  wrd2ind  11419  prmind2  12821  gropd  16059  grstructd2dom  16060  bj-intabssel  16578  bj-bdfindes  16736  bj-findes  16768