Theorem sbralieALT 3350

Description: Alternative shorter proof of sbralie 3349 dependent on ax-ext 2741, df-cleq 2761, df-clel 2844. (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 3322	. . 3 ⊢ (∀𝑦 ∈ 𝑥 𝜓 ↔ ∀𝑧 ∈ 𝑥 [𝑧 / 𝑦]𝜓)
2	1	sbbii 2116	. 2 ⊢ ([𝑦 / 𝑥]∀𝑦 ∈ 𝑥 𝜓 ↔ [𝑦 / 𝑥]∀𝑧 ∈ 𝑥 [𝑧 / 𝑦]𝜓)
3		raleq 3326	. . 3 ⊢ (𝑥 = 𝑦 → (∀𝑧 ∈ 𝑥 [𝑧 / 𝑦]𝜓 ↔ ∀𝑧 ∈ 𝑦 [𝑧 / 𝑦]𝜓))
4	3	sbievw 2134	. 2 ⊢ ([𝑦 / 𝑥]∀𝑧 ∈ 𝑥 [𝑧 / 𝑦]𝜓 ↔ ∀𝑧 ∈ 𝑦 [𝑧 / 𝑦]𝜓)
5		cbvralsvw 3322	. . 3 ⊢ (∀𝑧 ∈ 𝑦 [𝑧 / 𝑦]𝜓 ↔ ∀𝑥 ∈ 𝑦 [𝑥 / 𝑧][𝑧 / 𝑦]𝜓)
6		sbco2vv 2140	. . . . 5 ⊢ ([𝑥 / 𝑧][𝑧 / 𝑦]𝜓 ↔ [𝑥 / 𝑦]𝜓)
7		sbralie.1	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
8	7	bicomd 226	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜑))
9	8	equcoms 2047	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝜓 ↔ 𝜑))
10	9	sbievw 2134	. . . . 5 ⊢ ([𝑥 / 𝑦]𝜓 ↔ 𝜑)
11	6, 10	bitri 278	. . . 4 ⊢ ([𝑥 / 𝑧][𝑧 / 𝑦]𝜓 ↔ 𝜑)
12	11	ralbii 3117	. . 3 ⊢ (∀𝑥 ∈ 𝑦 [𝑥 / 𝑧][𝑧 / 𝑦]𝜓 ↔ ∀𝑥 ∈ 𝑦 𝜑)
13	5, 12	bitri 278	. 2 ⊢ (∀𝑧 ∈ 𝑦 [𝑧 / 𝑦]𝜓 ↔ ∀𝑥 ∈ 𝑦 𝜑)
14	2, 4, 13	3bitrri 301	1 ⊢ (∀𝑥 ∈ 𝑦 𝜑 ↔ [𝑦 / 𝑥]∀𝑦 ∈ 𝑥 𝜓)

Syntax hints:   → wi 4   ↔ wb 209  [wsb 2097  ∀wral 3085

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-11 2198  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-sb 2098  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096

This theorem is referenced by: (None)