Theorem sbralieALT 3331

Description: Alternative shorter proof of sbralie 3330 dependent on ax-ext 2724, df-cleq 2744, df-clel 2827. (Contributed by NM, 5-Sep-2004.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

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

Assertion

Ref	Expression
sbralieALT	⊢ (∀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 sbralieALT

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvralsvw 3303	. . 3 ⊢ (∀𝑦 ∈ 𝑥 𝜓 ↔ ∀𝑧 ∈ 𝑥 [𝑧 / 𝑦]𝜓)
2	1	sbbii 2099	. 2 ⊢ ([𝑦 / 𝑥]∀𝑦 ∈ 𝑥 𝜓 ↔ [𝑦 / 𝑥]∀𝑧 ∈ 𝑥 [𝑧 / 𝑦]𝜓)
3		raleq 3307	. . 3 ⊢ (𝑥 = 𝑦 → (∀𝑧 ∈ 𝑥 [𝑧 / 𝑦]𝜓 ↔ ∀𝑧 ∈ 𝑦 [𝑧 / 𝑦]𝜓))
4	3	sbievw 2117	. 2 ⊢ ([𝑦 / 𝑥]∀𝑧 ∈ 𝑥 [𝑧 / 𝑦]𝜓 ↔ ∀𝑧 ∈ 𝑦 [𝑧 / 𝑦]𝜓)
5		cbvralsvw 3303	. . 3 ⊢ (∀𝑧 ∈ 𝑦 [𝑧 / 𝑦]𝜓 ↔ ∀𝑥 ∈ 𝑦 [𝑥 / 𝑧][𝑧 / 𝑦]𝜓)
6		sbco2vv 2123	. . . . 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 2030	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝜓 ↔ 𝜑))
10	9	sbievw 2117	. . . . 5 ⊢ ([𝑥 / 𝑦]𝜓 ↔ 𝜑)
11	6, 10	bitri 277	. . . 4 ⊢ ([𝑥 / 𝑧][𝑧 / 𝑦]𝜓 ↔ 𝜑)
12	11	ralbii 3098	. . 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 2080  ∀wral 3066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-11 2181  ax-ext 2724

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1790  df-sb 2081  df-cleq 2744  df-clel 2827  df-ral 3067  df-rex 3077

This theorem is referenced by: (None)