Theorem sbcbig 3839

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 3790	. 2 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥](𝜑 ↔ 𝜓) ↔ [𝐴 / 𝑥](𝜑 ↔ 𝜓)))
2		dfsbcq2 3790	. . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
3		dfsbcq2 3790	. . 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 3556	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 ↔ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1539  [wsb 2063   ∈ wcel 2107  [wsbc 3787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-sbc 3788

This theorem is referenced by:  sbcbi1  3846  sbcabel  3877  opsbc2ie  32496  bnj89  34736  bj-sbeq  36903  bj-sbceqgALT  36904  sbcbi  44564  sbc3orgVD  44876  sbcbiVD  44901  modelaxreplem3  45002