Theorem sbceq1a 2999

Description: Equality theorem for class substitution. Class version of sbequ12 1785. (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 1788	. 2 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑)
2		dfsbcq2 2992	. 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 1364  [wsb 1776  [wsbc 2989

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-sbc 2990

This theorem is referenced by:  sbceq2a  3000  elrabsf  3028  cbvralcsf  3147  cbvrexcsf  3148  euotd  4288  omsinds  4659  elfvmptrab1  5659  ralrnmpt  5707  rexrnmpt  5708  riotass2  5907  riotass  5908  elovmporab  6127  elovmporab1w  6128  uchoice  6204  sbcopeq1a  6254  mpoxopoveq  6307  findcard2  6959  findcard2s  6960  ac6sfi  6968  opabfi  7008  dcfi  7056  indpi  7426  nn0ind-raph  9460  indstr  9684  fzrevral  10197  exfzdc  10333  zsupcllemstep  10336  infssuzex  10340  uzsinds  10553  prmind2  12313  bj-intabssel  15519  bj-bdfindes  15679  bj-findes  15711