Theorem sbceq1a 2974

Description: Equality theorem for class substitution. Class version of sbequ12 1771. (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 1774	. 2 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑)
2		dfsbcq2 2967	. 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 1353  [wsb 1762  [wsbc 2964

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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-sbc 2965

This theorem is referenced by:  sbceq2a  2975  elrabsf  3003  cbvralcsf  3121  cbvrexcsf  3122  euotd  4256  omsinds  4623  elfvmptrab1  5612  ralrnmpt  5660  rexrnmpt  5661  riotass2  5859  riotass  5860  sbcopeq1a  6190  mpoxopoveq  6243  findcard2  6891  findcard2s  6892  ac6sfi  6900  dcfi  6982  indpi  7343  nn0ind-raph  9372  indstr  9595  fzrevral  10107  exfzdc  10242  uzsinds  10444  zsupcllemstep  11948  infssuzex  11952  prmind2  12122  bj-intabssel  14626  bj-bdfindes  14786  bj-findes  14818