Theorem bnj538OLD 30810

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) Obsolete version of bnj538 30809 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 2917	. . 3 ⊢ (∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜑))
2	1	sbcbii 3491	. 2 ⊢ ([𝐴 / 𝑦]∀𝑥 ∈ 𝐵 𝜑 ↔ [𝐴 / 𝑦]∀𝑥(𝑥 ∈ 𝐵 → 𝜑))
3		bnj538OLD.1	. . . . . 6 ⊢ 𝐴 ∈ V
4		sbcimg 3477	. . . . . 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 30807	. . . . . 6 ⊢ ([𝐴 / 𝑦]𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵)
7	6	imbi1i 339	. . . . 5 ⊢ (([𝐴 / 𝑦]𝑥 ∈ 𝐵 → [𝐴 / 𝑦]𝜑) ↔ (𝑥 ∈ 𝐵 → [𝐴 / 𝑦]𝜑))
8	5, 7	bitri 264	. . . 4 ⊢ ([𝐴 / 𝑦](𝑥 ∈ 𝐵 → 𝜑) ↔ (𝑥 ∈ 𝐵 → [𝐴 / 𝑦]𝜑))
9	8	albii 1747	. . 3 ⊢ (∀𝑥[𝐴 / 𝑦](𝑥 ∈ 𝐵 → 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐵 → [𝐴 / 𝑦]𝜑))
10		sbcal 3485	. . 3 ⊢ ([𝐴 / 𝑦]∀𝑥(𝑥 ∈ 𝐵 → 𝜑) ↔ ∀𝑥[𝐴 / 𝑦](𝑥 ∈ 𝐵 → 𝜑))
11		df-ral 2917	. . 3 ⊢ (∀𝑥 ∈ 𝐵 [𝐴 / 𝑦]𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐵 → [𝐴 / 𝑦]𝜑))
12	9, 10, 11	3bitr4i 292	. 2 ⊢ ([𝐴 / 𝑦]∀𝑥(𝑥 ∈ 𝐵 → 𝜑) ↔ ∀𝑥 ∈ 𝐵 [𝐴 / 𝑦]𝜑)
13	2, 12	bitri 264	1 ⊢ ([𝐴 / 𝑦]∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ 𝐵 [𝐴 / 𝑦]𝜑)

Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1481   ∈ wcel 1990  ∀wral 2912  Vcvv 3200  [wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-ral 2917  df-v 3202  df-sbc 3436

This theorem is referenced by: (None)