Theorem sbcbig 3846

Description: Distribution of class substitution over biconditional. (Contributed by Raph Levien, 10-Apr-2004.)

Assertion

Ref	Expression
sbcbig	⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 ↔ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)))

Proof of Theorem sbcbig

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 3794	. 2 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥](𝜑 ↔ 𝜓) ↔ [𝐴 / 𝑥](𝜑 ↔ 𝜓)))
2		dfsbcq2 3794	. . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
3		dfsbcq2 3794	. . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜓 ↔ [𝐴 / 𝑥]𝜓))
4	2, 3	bibi12d 345	. 2 ⊢ (𝑦 = 𝐴 → (([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓) ↔ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)))
5		sbbi 2307	. 2 ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓))
6	1, 4, 5	vtoclbg 3557	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 ↔ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537  [wsb 2062   ∈ wcel 2106  [wsbc 3791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-sbc 3792

This theorem is referenced by:  sbcbi1  3853  sbcabel  3887  opsbc2ie  32504  bnj89  34714  bj-sbeq  36884  bj-sbceqgALT  36885  sbcbi  44537  sbc3orgVD  44849  sbcbiVD  44874  modelaxreplem3  44945