Theorem sbcbig 3798

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 3750	. 2 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥](𝜑 ↔ 𝜓) ↔ [𝐴 / 𝑥](𝜑 ↔ 𝜓)))
2		dfsbcq2 3750	. . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
3		dfsbcq2 3750	. . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜓 ↔ [𝐴 / 𝑥]𝜓))
4	2, 3	bibi12d 348	. 2 ⊢ (𝑦 = 𝐴 → (([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓) ↔ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)))
5		sbbi 2345	. 2 ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓))
6	1, 4, 5	vtoclbg 3527	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 ↔ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  [wsb 2099   ∈ wcel 2146  [wsbc 3747

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-12 2216  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-sbc 3748

This theorem is used by:  sbcbi1  3804  sbcabel  3834  opsbc2ie  32859  bnj89  35142  bj-sbeq  37577  bj-sbceqgALT  37578  sbcbi  45289  sbc3orgVD  45600  sbcbiVD  45625  modelaxreplem3  45730