Theorem sbsbc 3049

Description: Show that df-sb 1812 and df-sbc 3046 are equivalent when the class term 𝐴 in df-sbc 3046 is a setvar variable. This theorem lets us reuse theorems based on df-sb 1812 for proofs involving df-sbc 3046. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.)

Assertion

Ref	Expression
sbsbc	⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Proof of Theorem sbsbc

Step	Hyp	Ref	Expression
1		eqid 2234	. 2 ⊢ 𝑦 = 𝑦
2		dfsbcq2 3048	. 2 ⊢ (𝑦 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
3	1, 2	ax-mp 5	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 105  [wsb 1811  [wsbc 3045

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-4 1559  ax-17 1575  ax-ial 1583  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-clab 2221  df-cleq 2227  df-clel 2230  df-sbc 3046

This theorem is referenced by:  spsbc  3057  sbcid  3061  sbcco  3067  sbcco2  3068  sbcie2g  3079  eqsbc1  3085  sbcralt  3122  sbcrext  3123  sbnfc2  3202  csbabg  3203  cbvralcsf  3204  cbvrexcsf  3205  cbvreucsf  3206  cbvrabcsf  3207  isarep1  5447  finexdc  7173  ssfirab  7210  zsupcllemstep  10611  bezoutlemmain  12719  bdsbc  16754