Theorem sbcbiVD 45699

Description: Implication form of sbcbii 3795. 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 45363 is sbcbiVD 45699 without virtual deductions and was automatically derived from sbcbiVD 45699.

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 45398	. . . 4 ⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   )
2		idn2 45437	. . . . 5 ⊢ (   𝐴 ∈ 𝐵   ,   ∀𝑥(𝜑 ↔ 𝜓)   ▶   ∀𝑥(𝜑 ↔ 𝜓)   )
3		spsbc 3752	. . . . 5 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥(𝜑 ↔ 𝜓) → [𝐴 / 𝑥](𝜑 ↔ 𝜓)))
4	1, 2, 3	e12 45547	. . . 4 ⊢ (   𝐴 ∈ 𝐵   ,   ∀𝑥(𝜑 ↔ 𝜓)   ▶   [𝐴 / 𝑥](𝜑 ↔ 𝜓)   )
5		sbcbig 3790	. . . . 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 45547	. . 3 ⊢ (   𝐴 ∈ 𝐵   ,   ∀𝑥(𝜑 ↔ 𝜓)   ▶   ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)   )
8	7	in2 45429	. 2 ⊢ (   𝐴 ∈ 𝐵   ▶   (∀𝑥(𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓))   )
9	8	in1 45395	1 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥(𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568   ∈ wcel 2145  [wsbc 3739

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 2147  ax-9 2155  ax-10 2178  ax-12 2213  ax-ext 2732

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 2739  df-cleq 2752  df-clel 2835  df-sbc 3740  df-vd1 45394  df-vd2 45402

This theorem is used by: (None)