Theorem sbcbi 44537

Description: Implication form of sbcbii 3852. sbcbi 44537 is sbcbiVD 44874 without virtual deductions and was automatically derived from sbcbiVD 44874 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem sbcbi

Step	Hyp	Ref	Expression
1		spsbc 3804	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝜑 ↔ 𝜓) → [𝐴 / 𝑥](𝜑 ↔ 𝜓)))
2		sbcbig 3846	. 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 ↔ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)))
3	1, 2	sylibd 239	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1535   ∈ 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:  trsbcVD  44875  sbcssgVD  44881