Theorem sbralie 3471

Description: Implicit to explicit substitution that swaps variables in a quantified expression. (Contributed by NM, 5-Sep-2004.)

Hypothesis

Ref	Expression
sbralie.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
sbralie	⊢ (∀𝑥 ∈ 𝑦 𝜑 ↔ [𝑦 / 𝑥]∀𝑦 ∈ 𝑥 𝜓)

Distinct variable groups:   𝑥,𝑦   𝜓,𝑥   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem sbralie

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvralsvw 3467	. . 3 ⊢ (∀𝑦 ∈ 𝑥 𝜓 ↔ ∀𝑧 ∈ 𝑥 [𝑧 / 𝑦]𝜓)
2	1	sbbii 2081	. 2 ⊢ ([𝑦 / 𝑥]∀𝑦 ∈ 𝑥 𝜓 ↔ [𝑦 / 𝑥]∀𝑧 ∈ 𝑥 [𝑧 / 𝑦]𝜓)
3		raleq 3405	. . 3 ⊢ (𝑥 = 𝑦 → (∀𝑧 ∈ 𝑥 [𝑧 / 𝑦]𝜓 ↔ ∀𝑧 ∈ 𝑦 [𝑧 / 𝑦]𝜓))
4	3	sbievw 2103	. 2 ⊢ ([𝑦 / 𝑥]∀𝑧 ∈ 𝑥 [𝑧 / 𝑦]𝜓 ↔ ∀𝑧 ∈ 𝑦 [𝑧 / 𝑦]𝜓)
5		cbvralsvw 3467	. . 3 ⊢ (∀𝑧 ∈ 𝑦 [𝑧 / 𝑦]𝜓 ↔ ∀𝑥 ∈ 𝑦 [𝑥 / 𝑧][𝑧 / 𝑦]𝜓)
6		sbco2vv 2108	. . . . 5 ⊢ ([𝑥 / 𝑧][𝑧 / 𝑦]𝜓 ↔ [𝑥 / 𝑦]𝜓)
7		sbralie.1	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
8	7	bicomd 225	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜑))
9	8	equcoms 2027	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝜓 ↔ 𝜑))
10	9	sbievw 2103	. . . . 5 ⊢ ([𝑥 / 𝑦]𝜓 ↔ 𝜑)
11	6, 10	bitri 277	. . . 4 ⊢ ([𝑥 / 𝑧][𝑧 / 𝑦]𝜓 ↔ 𝜑)
12	11	ralbii 3165	. . 3 ⊢ (∀𝑥 ∈ 𝑦 [𝑥 / 𝑧][𝑧 / 𝑦]𝜓 ↔ ∀𝑥 ∈ 𝑦 𝜑)
13	5, 12	bitri 277	. 2 ⊢ (∀𝑧 ∈ 𝑦 [𝑧 / 𝑦]𝜓 ↔ ∀𝑥 ∈ 𝑦 𝜑)
14	2, 4, 13	3bitrri 300	1 ⊢ (∀𝑥 ∈ 𝑦 𝜑 ↔ [𝑦 / 𝑥]∀𝑦 ∈ 𝑥 𝜓)

Syntax hints:   → wi 4   ↔ wb 208  [wsb 2069  ∀wral 3138

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785  df-sb 2070  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143

This theorem is referenced by:  tfinds2  7578