Theorem cbvex-P7 1066

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

Hypotheses

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

Assertion

Ref	Expression
cbvex-P7	⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Distinct variable group:   𝑥,𝑦

Proof of Theorem cbvex-P7

Step	Hyp	Ref	Expression
1		cbvex-P7.2	. . 3 ⊢ Ⅎ𝑦𝜑
2		cbvex-P7.1	. . 3 ⊢ Ⅎ𝑥𝜓
3		cbvex-P7.3	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	3	bisym-P3.33b 298	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜑))
5		eqsym-P7.CL.SYM 938	. . . 4 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
6	4, 5	subiml2-P4.RC 541	. . 3 ⊢ (𝑦 = 𝑥 → (𝜓 ↔ 𝜑))
7	1, 2, 6	cbvex-P7-L1 1065	. 2 ⊢ (∃𝑥𝜑 → ∃𝑦𝜓)
8	2, 1, 3	cbvex-P7-L1 1065	. 2 ⊢ (∃𝑦𝜓 → ∃𝑥𝜑)
9	7, 8	rcp-NDBII0 239	1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Syntax hints:  term-obj 1   = wff-equals 6   → wff-imp 10   ↔ wff-bi 104  ∃wff-exists 595  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:  cbvex-P7.VR1of2  1067  cbvex-P7.VR2of2  1068  cbvex-P7.VR12of2  1069  cbvexpsub-P7  1072