Theorem cbvrmov 2729

Description: Change the bound variable of a restricted at-most-one quantifier using implicit substitution. (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Hypothesis

Ref	Expression
cbvralv.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvrmov	⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓)

Distinct variable groups:   𝑥,𝐴   𝑦,𝐴   𝜑,𝑦   𝜓,𝑥

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

Proof of Theorem cbvrmov

Step	Hyp	Ref	Expression
1		nfv 1539	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1539	. 2 ⊢ Ⅎ𝑥𝜓
3		cbvralv.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	1, 2, 3	cbvrmo 2725	1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ↔ wb 105  ∃*wrmo 2475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-reu 2479  df-rmo 2480

This theorem is referenced by: (None)