Theorem cbvall-P6 751

Description: Change of Bound Variable Law for '∀𝑥' (non-freeness condition).

Hypotheses

Ref	Expression
cbvall-P6.1	⊢ Ⅎ𝑥𝜓
cbvall-P6.2	⊢ Ⅎ𝑦𝜑
cbvall-P6.3	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvall-P6	⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Distinct variable group:   𝑥,𝑦

Proof of Theorem cbvall-P6

Step	Hyp	Ref	Expression
1		cbvall-P6.1	. . 3 ⊢ Ⅎ𝑥𝜓
2		cbvall-P6.2	. . 3 ⊢ Ⅎ𝑦𝜑
3		cbvall-P6.3	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	3	ndbief-P3.14 179	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
5	1, 2, 4	cbvall-P6-L1 750	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)
6	3	ndbier-P3.15 180	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜓 → 𝜑))
7		eqsym-P5.CL.SYM 629	. . . 4 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
8	6, 7	subiml2-P4.RC 541	. . 3 ⊢ (𝑦 = 𝑥 → (𝜓 → 𝜑))
9	2, 1, 8	cbvall-P6-L1 750	. 2 ⊢ (∀𝑦𝜓 → ∀𝑥𝜑)
10	5, 9	rcp-NDBII0 239	1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Syntax hints:  term-obj 1   = wff-equals 6  ∀wff-forall 8   → wff-imp 10   ↔ wff-bi 104  Ⅎwff-nfree 681

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14  ax-GEN 15  ax-L4 16  ax-L5 17  ax-L6 18  ax-L7 19  ax-L10 27  ax-L11 28  ax-L12 29

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133  df-or-D2.3 145  df-true-D2.4 155  df-rcp-AND3 161  df-exists-D5.1 596  df-nfree-D6.1 682

This theorem is referenced by:  cbvex-P6  752  cbvallpsub-P6  768