Theorem cbvall-P7 1061

Description: Change of Bound Variable Theorem for '∀𝑥'. †

Hypotheses

Ref	Expression
cbvall-P7.1	⊢ Ⅎ𝑥𝜓
cbvall-P7.2	⊢ Ⅎ𝑦𝜑
cbvall-P7.3	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvall-P7	⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Distinct variable group:   𝑥,𝑦

Proof of Theorem cbvall-P7

Step	Hyp	Ref	Expression
1		cbvall-P7.1	. . 3 ⊢ Ⅎ𝑥𝜓
2		cbvall-P7.2	. . 3 ⊢ Ⅎ𝑦𝜑
3		cbvall-P7.3	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	1, 2, 3	cbvall-P7-L1 1060	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)
5	3	bisym-P3.33b 298	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜑))
6		eqsym-P7.CL.SYM 938	. . . 4 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
7	5, 6	subiml2-P4.RC 541	. . 3 ⊢ (𝑦 = 𝑥 → (𝜓 ↔ 𝜑))
8	2, 1, 7	cbvall-P7-L1 1060	. 2 ⊢ (∀𝑦𝜓 → ∀𝑥𝜑)
9	4, 8	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  df-psub-D6.2 716

This theorem is referenced by:  cbvall-P7.VR1of2  1062  cbvall-P7.VR2of2  1063  cbvall-P7.VR12of2  1064  cbvallpsub-P7  1070