Theorem bnj538OLD 30064

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) Obsolete version of bnj538 30063 as of 30-Mar-2020. (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj538OLD.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
bnj538OLD	⊢ ([𝐴 / 𝑦]∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ 𝐵 [𝐴 / 𝑦]𝜑)

Distinct variable groups:   𝑥,𝐴   𝑦,𝐵   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑦)   𝐵(𝑥)

Proof of Theorem bnj538OLD

Step	Hyp	Ref	Expression
1		df-ral 2901	. . 3 ⊢ (∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜑))
2	1	sbcbii 3458	. 2 ⊢ ([𝐴 / 𝑦]∀𝑥 ∈ 𝐵 𝜑 ↔ [𝐴 / 𝑦]∀𝑥(𝑥 ∈ 𝐵 → 𝜑))
3		bnj538OLD.1	. . . . . 6 ⊢ 𝐴 ∈ V
4		sbcimg 3444	. . . . . 6 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑦](𝑥 ∈ 𝐵 → 𝜑) ↔ ([𝐴 / 𝑦]𝑥 ∈ 𝐵 → [𝐴 / 𝑦]𝜑)))
5	3, 4	ax-mp 5	. . . . 5 ⊢ ([𝐴 / 𝑦](𝑥 ∈ 𝐵 → 𝜑) ↔ ([𝐴 / 𝑦]𝑥 ∈ 𝐵 → [𝐴 / 𝑦]𝜑))
6	3	bnj525 30061	. . . . . 6 ⊢ ([𝐴 / 𝑦]𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵)
7	6	imbi1i 338	. . . . 5 ⊢ (([𝐴 / 𝑦]𝑥 ∈ 𝐵 → [𝐴 / 𝑦]𝜑) ↔ (𝑥 ∈ 𝐵 → [𝐴 / 𝑦]𝜑))
8	5, 7	bitri 263	. . . 4 ⊢ ([𝐴 / 𝑦](𝑥 ∈ 𝐵 → 𝜑) ↔ (𝑥 ∈ 𝐵 → [𝐴 / 𝑦]𝜑))
9	8	albii 1737	. . 3 ⊢ (∀𝑥[𝐴 / 𝑦](𝑥 ∈ 𝐵 → 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐵 → [𝐴 / 𝑦]𝜑))
10		sbcal 3452	. . 3 ⊢ ([𝐴 / 𝑦]∀𝑥(𝑥 ∈ 𝐵 → 𝜑) ↔ ∀𝑥[𝐴 / 𝑦](𝑥 ∈ 𝐵 → 𝜑))
11		df-ral 2901	. . 3 ⊢ (∀𝑥 ∈ 𝐵 [𝐴 / 𝑦]𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐵 → [𝐴 / 𝑦]𝜑))
12	9, 10, 11	3bitr4i 291	. 2 ⊢ ([𝐴 / 𝑦]∀𝑥(𝑥 ∈ 𝐵 → 𝜑) ↔ ∀𝑥 ∈ 𝐵 [𝐴 / 𝑦]𝜑)
13	2, 12	bitri 263	1 ⊢ ([𝐴 / 𝑦]∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ 𝐵 [𝐴 / 𝑦]𝜑)

Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  [wsbc 3402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-ral 2901  df-v 3175  df-sbc 3403

This theorem is referenced by: (None)