Theorem sbcbiVD 42110

Description: Implication form of sbcbii 3743. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sbcbi 41773 is sbcbiVD 42110 without virtual deductions and was automatically derived from sbcbiVD 42110.

1::	⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   )
2::	⊢ (   𝐴 ∈ 𝐵   ,   ∀𝑥(𝜑 ↔ 𝜓)    ▶   ∀𝑥(𝜑 ↔ 𝜓)   )
3:1,2:	⊢ (   𝐴 ∈ 𝐵   ,   ∀𝑥(𝜑 ↔ 𝜓)    ▶   [𝐴 / 𝑥](𝜑 ↔ 𝜓)   )
4:1,3:	⊢ (   𝐴 ∈ 𝐵   ,   ∀𝑥(𝜑 ↔ 𝜓)    ▶   ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)   )
5:4:	⊢ (   𝐴 ∈ 𝐵   ▶   (∀𝑥(𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓))   )
qed:5:	⊢ (𝐴 ∈ 𝐵 → (∀𝑥(𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)))

(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem sbcbiVD

Step	Hyp	Ref	Expression
1		idn1 41808	. . . 4 ⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   )
2		idn2 41847	. . . . 5 ⊢ (   𝐴 ∈ 𝐵   ,   ∀𝑥(𝜑 ↔ 𝜓)   ▶   ∀𝑥(𝜑 ↔ 𝜓)   )
3		spsbc 3696	. . . . 5 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥(𝜑 ↔ 𝜓) → [𝐴 / 𝑥](𝜑 ↔ 𝜓)))
4	1, 2, 3	e12 41958	. . . 4 ⊢ (   𝐴 ∈ 𝐵   ,   ∀𝑥(𝜑 ↔ 𝜓)   ▶   [𝐴 / 𝑥](𝜑 ↔ 𝜓)   )
5		sbcbig 3737	. . . . 5 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥](𝜑 ↔ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)))
6	5	biimpd 232	. . . 4 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥](𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)))
7	1, 4, 6	e12 41958	. . 3 ⊢ (   𝐴 ∈ 𝐵   ,   ∀𝑥(𝜑 ↔ 𝜓)   ▶   ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)   )
8	7	in2 41839	. 2 ⊢ (   𝐴 ∈ 𝐵   ▶   (∀𝑥(𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓))   )
9	8	in1 41805	1 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥(𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1541   ∈ wcel 2112  [wsbc 3683

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-sbc 3684  df-vd1 41804  df-vd2 41812

This theorem is referenced by: (None)