Theorem cbvmowOLD 2624

Description: Obsolete version of cbvmow 2623 as of 23-May-2024. (Contributed by NM, 9-Mar-1995.) (Revised by Gino Giotto, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
cbvmowOLD	⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem cbvmowOLD

Step	Hyp	Ref	Expression
1		cbvmowOLD.1	. . . . 5 ⊢ Ⅎ𝑦𝜑
2	1	sb8ev 2364	. . . 4 ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
3	1	sb8euv 2620	. . . 4 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
4	2, 3	imbi12i 355	. . 3 ⊢ ((∃𝑥𝜑 → ∃!𝑥𝜑) ↔ (∃𝑦[𝑦 / 𝑥]𝜑 → ∃!𝑦[𝑦 / 𝑥]𝜑))
5		moeu 2603	. . 3 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
6		moeu 2603	. . 3 ⊢ (∃*𝑦[𝑦 / 𝑥]𝜑 ↔ (∃𝑦[𝑦 / 𝑥]𝜑 → ∃!𝑦[𝑦 / 𝑥]𝜑))
7	4, 5, 6	3bitr4i 307	. 2 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑)
8		cbvmowOLD.2	. . . 4 ⊢ Ⅎ𝑥𝜓
9		cbvmowOLD.3	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
10	8, 9	sbiev 2323	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
11	10	mobii 2566	. 2 ⊢ (∃*𝑦[𝑦 / 𝑥]𝜑 ↔ ∃*𝑦𝜓)
12	7, 11	bitri 278	1 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 209  ∃wex 1782  Ⅎwnf 1786  [wsb 2070  ∃*wmo 2556  ∃!weu 2588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-10 2143  ax-11 2159  ax-12 2176

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589

This theorem is referenced by: (None)