Theorem cbvexpsub-P7 1072

Description: Change of Bound Variable for '∃𝑥' with Proper Substitution. †

Hypothesis

Ref	Expression
cbvexpsub-P7.1	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
cbvexpsub-P7	⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)

Distinct variable group:   𝑥,𝑦

Proof of Theorem cbvexpsub-P7

Step	Hyp	Ref	Expression
1		ndpsub3-P7.15 840	. 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
2		cbvexpsub-P7.1	. 2 ⊢ Ⅎ𝑦𝜑
3		ndpsub2-P7.14 839	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
4	1, 2, 3	cbvex-P7 1066	1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)

Syntax hints:  term-obj 1   ↔ wff-bi 104  ∃wff-exists 595  Ⅎwff-nfree 681  [wff-psub 714

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:  cbvexpsub-P7.VR  1073